From 38b4d78db3ad30a5dd1189b63f3830ff4c06e6e7 Mon Sep 17 00:00:00 2001 From: andatoshiki <101481353+andatoshiki@users.noreply.github.com> Date: Tue, 23 May 2023 16:38:27 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20andatosh?= =?UTF-8?q?iki/toshiki-notebook@edb86108d70ffb4481b2da47afbae4424acdd129?= =?UTF-8?q?=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 404.html | 6 +- academic/chemistry/index.html | 10 +- academic/chemistry/notes/12-5.html | 10 +- academic/chemistry/problems/02-20.html | 10 +- academic/chemistry/problems/03-02-1.html | 10 +- academic/chemistry/problems/03-02-2.html | 10 +- academic/chemistry/problems/03-02-3.html | 10 +- academic/literature/index.html | 10 +- .../writing/methods-of-development.html | 10 +- academic/physics/index.html | 10 +- academic/physics/ipho-formulas-jpn/1.html | 10 +- academic/physics/ipho-formulas-jpn/10.html | 10 +- academic/physics/ipho-formulas-jpn/11.html | 10 +- academic/physics/ipho-formulas-jpn/12.html | 10 +- academic/physics/ipho-formulas-jpn/13.html | 10 +- academic/physics/ipho-formulas-jpn/2.html | 10 +- academic/physics/ipho-formulas-jpn/3.html | 10 +- academic/physics/ipho-formulas-jpn/4.html | 10 +- academic/physics/ipho-formulas-jpn/5.html | 10 +- academic/physics/ipho-formulas-jpn/6.html | 10 +- academic/physics/ipho-formulas-jpn/7.html | 10 +- academic/physics/ipho-formulas-jpn/8.html | 10 +- academic/physics/ipho-formulas-jpn/9.html | 10 +- academic/vocabulary/2023/02/2023-02-27.html | 10 +- academic/vocabulary/index.html | 10 +- application/markdown-it-katex/how-to-use.html | 16 +-- .../markdown-it-katex/support-function.html | 10 +- .../markdown-it-katex/support-table.html | 10 +- application/markdown-it-katex/tips.html | 12 +- ...> academic_chemistry_index.md.20e560a6.js} | 2 +- ...demic_chemistry_index.md.20e560a6.lean.js} | 2 +- ...demic_chemistry_notes_12-5.md.7fe55c03.js} | 2 +- ..._chemistry_notes_12-5.md.7fe55c03.lean.js} | 2 +- ...c_chemistry_problems_02-20.md.6c822dc9.js} | 2 +- ...mistry_problems_02-20.md.6c822dc9.lean.js} | 2 +- ...chemistry_problems_03-02-1.md.765a5cd1.js} | 2 +- ...stry_problems_03-02-1.md.765a5cd1.lean.js} | 2 +- ...chemistry_problems_03-02-2.md.35916f88.js} | 2 +- ...stry_problems_03-02-2.md.35916f88.lean.js} | 2 +- ...chemistry_problems_03-02-3.md.2d09c751.js} | 2 +- ...stry_problems_03-02-3.md.2d09c751.lean.js} | 2 +- ... academic_literature_index.md.b9024d79.js} | 2 +- ...emic_literature_index.md.b9024d79.lean.js} | 2 +- ...ing_methods-of-development.md.b79948e6.js} | 2 +- ...ethods-of-development.md.b79948e6.lean.js} | 2 +- ... => academic_physics_index.md.837a8425.js} | 2 +- ...cademic_physics_index.md.837a8425.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_1.md.1be1e0d1.js} | 2 +- ...s_ipho-formulas-jpn_1.md.1be1e0d1.lean.js} | 2 +- ...ysics_ipho-formulas-jpn_10.md.53acc7b0.js} | 2 +- ..._ipho-formulas-jpn_10.md.53acc7b0.lean.js} | 2 +- ...ysics_ipho-formulas-jpn_11.md.80185913.js} | 2 +- ..._ipho-formulas-jpn_11.md.80185913.lean.js} | 2 +- ...ysics_ipho-formulas-jpn_12.md.bff53a1c.js} | 2 +- ..._ipho-formulas-jpn_12.md.bff53a1c.lean.js} | 2 +- ...ysics_ipho-formulas-jpn_13.md.40dc3c65.js} | 2 +- ..._ipho-formulas-jpn_13.md.40dc3c65.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_2.md.2b6166b8.js} | 2 +- ...s_ipho-formulas-jpn_2.md.2b6166b8.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_3.md.24fb7253.js} | 2 +- ...s_ipho-formulas-jpn_3.md.24fb7253.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_4.md.6ae94d2b.js} | 2 +- ...s_ipho-formulas-jpn_4.md.6ae94d2b.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_5.md.db50d685.js} | 2 +- ...s_ipho-formulas-jpn_5.md.db50d685.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_6.md.2e655de1.js} | 2 +- ...s_ipho-formulas-jpn_6.md.2e655de1.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_7.md.7be7b512.js} | 2 +- ...s_ipho-formulas-jpn_7.md.7be7b512.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_8.md.02aeb40e.js} | 2 +- ...s_ipho-formulas-jpn_8.md.02aeb40e.lean.js} | 2 +- ...hysics_ipho-formulas-jpn_9.md.fc474d28.js} | 2 +- ...s_ipho-formulas-jpn_9.md.fc474d28.lean.js} | 2 +- ...abulary_2023_02_2023-02-27.md.50b28361.js} | 2 +- ...ry_2023_02_2023-02-27.md.50b28361.lean.js} | 2 +- ... academic_vocabulary_index.md.b0a1288c.js} | 2 +- ...emic_vocabulary_index.md.b0a1288c.lean.js} | 2 +- ...rkdown-it-katex_how-to-use.md.b720171d.js} | 8 +- ...n-it-katex_how-to-use.md.b720171d.lean.js} | 4 +- ...-it-katex_support-function.md.1d1d4129.js} | 2 +- ...atex_support-function.md.1d1d4129.lean.js} | 2 +- ...own-it-katex_support-table.md.646afbf1.js} | 2 +- ...t-katex_support-table.md.646afbf1.lean.js} | 2 +- ...ion_markdown-it-katex_tips.md.2d4f69d7.js} | 2 +- ...arkdown-it-katex_tips.md.2d4f69d7.lean.js} | 2 +- ... => development_aws_README.md.ea2d6a9b.js} | 2 +- ...evelopment_aws_README.md.ea2d6a9b.lean.js} | 2 +- ...opment_aws_acknowledgement.md.2c27a486.js} | 2 +- ...t_aws_acknowledgement.md.2c27a486.lean.js} | 2 +- ...> development_aws_appendix.md.7272d46f.js} | 2 +- ...elopment_aws_appendix.md.7272d46f.lean.js} | 2 +- ...evelopment_aws_assignments.md.27524939.js} | 2 +- ...pment_aws_assignments.md.27524939.lean.js} | 2 +- ... => development_aws_author.md.a50d2f4c.js} | 2 +- ...evelopment_aws_author.md.a50d2f4c.lean.js} | 2 +- ... development_aws_aws-batch.md.2225dbd4.js} | 2 +- ...lopment_aws_aws-batch.md.2225dbd4.lean.js} | 2 +- ...opment_aws_aws-get-started.md.305acfd1.js} | 2 +- ...t_aws_aws-get-started.md.305acfd1.lean.js} | 2 +- ...=> development_aws_closing.md.73173906.js} | 2 +- ...velopment_aws_closing.md.73173906.lean.js} | 2 +- ...s => development_aws_cloud.md.d6418ba0.js} | 2 +- ...development_aws_cloud.md.d6418ba0.lean.js} | 2 +- ...elopment_aws_docker-system.md.b4b42fe0.js} | 2 +- ...ent_aws_docker-system.md.b4b42fe0.lean.js} | 2 +- ...ent_aws_handson-bashoutter.md.91af8955.js} | 2 +- ...ws_handson-bashoutter.md.91af8955.lean.js} | 2 +- ...evelopment_aws_handson-ec2.md.d527a33c.js} | 2 +- ...pment_aws_handson-ec2.md.d527a33c.lean.js} | 2 +- ...opment_aws_handson-jupyter.md.887c5eff.js} | 2 +- ...t_aws_handson-jupyter.md.887c5eff.lean.js} | 2 +- ...elopment_aws_handson-qabot.md.b244b90e.js} | 2 +- ...ent_aws_handson-qabot.md.b244b90e.lean.js} | 2 +- ...ent_aws_handson-serverless.md.a47ced9b.js} | 2 +- ...ws_handson-serverless.md.a47ced9b.lean.js} | 2 +- ...velopment_aws_introduction.md.a28e3444.js} | 2 +- ...ment_aws_introduction.md.a28e3444.lean.js} | 2 +- ...=> development_aws_license.md.3681aae7.js} | 2 +- ...velopment_aws_license.md.3681aae7.lean.js} | 2 +- ...js => development_aws_main.md.cc9bca88.js} | 2 +- ... development_aws_main.md.cc9bca88.lean.js} | 2 +- ...t_aws_scientific-computing.md.44b26ae4.js} | 2 +- ..._scientific-computing.md.44b26ae4.lean.js} | 2 +- ...development_aws_serverless.md.954dd7a1.js} | 2 +- ...opment_aws_serverless.md.954dd7a1.lean.js} | 2 +- ... development_aws_webserver.md.a8b089c4.js} | 2 +- ...lopment_aws_webserver.md.a8b089c4.lean.js} | 2 +- ...ment_file-naming-convention.md.404e82de.js | 67 +++++++++++ ...file-naming-convention.md.404e82de.lean.js | 1 + .../development_rclone-for-r2.md.e7f3a9b7.js | 55 +++++++++ ...elopment_rclone-for-r2.md.e7f3a9b7.lean.js | 1 + ...b99e.js => getting-started.md.ad0edf0e.js} | 2 +- ...js => getting-started.md.ad0edf0e.lean.js} | 2 +- ...ex.md.6f6f33ad.js => index.md.4559d5f1.js} | 2 +- ...33ad.lean.js => index.md.4559d5f1.lean.js} | 2 +- ... => javascript_notes_1_1-1.md.7c957dab.js} | 2 +- ...avascript_notes_1_1-1.md.7c957dab.lean.js} | 2 +- ... => javascript_notes_1_1-2.md.46f1230f.js} | 2 +- ...avascript_notes_1_1-2.md.46f1230f.lean.js} | 2 +- ....md.7ba79349.js => roadmap.md.a42ebae3.js} | 2 +- ...49.lean.js => roadmap.md.a42ebae3.lean.js} | 2 +- ...8.js => save_reading_index.md.983fe09d.js} | 2 +- ...=> save_reading_index.md.983fe09d.lean.js} | 2 +- ...=> save_reading_outliers_1.md.e3cd2c6d.js} | 2 +- ...ve_reading_outliers_1.md.e3cd2c6d.lean.js} | 2 +- ...=> save_reading_outliers_2.md.ad1783d7.js} | 2 +- ...ve_reading_outliers_2.md.ad1783d7.lean.js} | 2 +- ...=> save_reading_outliers_3.md.e6fcee78.js} | 2 +- ...ve_reading_outliers_3.md.e6fcee78.lean.js} | 2 +- ...=> save_reading_outliers_4.md.62591abe.js} | 2 +- ...ve_reading_outliers_4.md.62591abe.lean.js} | 2 +- ...{style.0b53ab5c.css => style.250ef51a.css} | 2 +- development/aws/README.html | 10 +- development/aws/acknowledgement.html | 10 +- development/aws/appendix.html | 10 +- development/aws/assignments.html | 10 +- development/aws/author.html | 10 +- development/aws/aws-batch.html | 10 +- development/aws/aws-get-started.html | 10 +- development/aws/closing.html | 10 +- development/aws/cloud.html | 10 +- development/aws/docker-system.html | 10 +- development/aws/handson-bashoutter.html | 10 +- development/aws/handson-ec2.html | 10 +- development/aws/handson-jupyter.html | 10 +- development/aws/handson-qabot.html | 10 +- development/aws/handson-serverless.html | 10 +- development/aws/introduction.html | 10 +- development/aws/license.html | 10 +- development/aws/main.html | 10 +- development/aws/scientific-computing.html | 10 +- development/aws/serverless.html | 10 +- development/aws/webserver.html | 10 +- development/file-naming-convention.html | 106 ++++++++++++++++++ development/rclone-for-r2.html | 94 ++++++++++++++++ getting-started.html | 10 +- hashmap.json | 2 +- index.html | 12 +- javascript/notes/1/1-1.html | 10 +- javascript/notes/1/1-2.html | 10 +- roadmap.html | 10 +- save/reading/index.html | 10 +- save/reading/outliers/1.html | 10 +- save/reading/outliers/2.html | 10 +- save/reading/outliers/3.html | 10 +- save/reading/outliers/4.html | 10 +- sitemap.xml | 2 +- 187 files changed, 752 insertions(+), 428 deletions(-) rename assets/{academic_chemistry_index.md.e22e1eca.js => academic_chemistry_index.md.20e560a6.js} (90%) rename assets/{academic_chemistry_index.md.e22e1eca.lean.js => academic_chemistry_index.md.20e560a6.lean.js} (90%) rename assets/{academic_chemistry_notes_12-5.md.fa35f18c.js => academic_chemistry_notes_12-5.md.7fe55c03.js} (99%) rename assets/{academic_chemistry_notes_12-5.md.fa35f18c.lean.js => academic_chemistry_notes_12-5.md.7fe55c03.lean.js} (99%) rename assets/{academic_chemistry_problems_02-20.md.e1fa1931.js => academic_chemistry_problems_02-20.md.6c822dc9.js} (99%) rename assets/{academic_chemistry_problems_02-20.md.e1fa1931.lean.js => academic_chemistry_problems_02-20.md.6c822dc9.lean.js} (99%) rename assets/{academic_chemistry_problems_03-02-1.md.3f33f487.js => academic_chemistry_problems_03-02-1.md.765a5cd1.js} (99%) rename assets/{academic_chemistry_problems_03-02-1.md.3f33f487.lean.js => academic_chemistry_problems_03-02-1.md.765a5cd1.lean.js} (99%) rename assets/{academic_chemistry_problems_03-02-2.md.bab6348d.js => academic_chemistry_problems_03-02-2.md.35916f88.js} (99%) rename assets/{academic_chemistry_problems_03-02-2.md.bab6348d.lean.js => academic_chemistry_problems_03-02-2.md.35916f88.lean.js} (99%) rename assets/{academic_chemistry_problems_03-02-3.md.90fcd861.js => academic_chemistry_problems_03-02-3.md.2d09c751.js} (99%) rename assets/{academic_chemistry_problems_03-02-3.md.90fcd861.lean.js => academic_chemistry_problems_03-02-3.md.2d09c751.lean.js} (99%) rename assets/{academic_literature_index.md.e8ed8386.js => academic_literature_index.md.b9024d79.js} (90%) rename assets/{academic_literature_index.md.e8ed8386.lean.js => academic_literature_index.md.b9024d79.lean.js} (90%) rename assets/{academic_literature_writing_methods-of-development.md.cb0d57b6.js => academic_literature_writing_methods-of-development.md.b79948e6.js} (99%) rename assets/{academic_literature_writing_methods-of-development.md.cb0d57b6.lean.js => academic_literature_writing_methods-of-development.md.b79948e6.lean.js} (91%) rename assets/{academic_physics_index.md.669c3eb1.js => academic_physics_index.md.837a8425.js} (90%) rename assets/{academic_physics_index.md.669c3eb1.lean.js => academic_physics_index.md.837a8425.lean.js} (90%) rename assets/{academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.js => academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.lean.js => academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_10.md.31925895.js => academic_physics_ipho-formulas-jpn_10.md.53acc7b0.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_10.md.31925895.lean.js => academic_physics_ipho-formulas-jpn_10.md.53acc7b0.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.js => academic_physics_ipho-formulas-jpn_11.md.80185913.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.lean.js => academic_physics_ipho-formulas-jpn_11.md.80185913.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_12.md.6b0b8711.js => academic_physics_ipho-formulas-jpn_12.md.bff53a1c.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_12.md.6b0b8711.lean.js => academic_physics_ipho-formulas-jpn_12.md.bff53a1c.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_13.md.b330f9cc.js => academic_physics_ipho-formulas-jpn_13.md.40dc3c65.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_13.md.b330f9cc.lean.js => academic_physics_ipho-formulas-jpn_13.md.40dc3c65.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.js => academic_physics_ipho-formulas-jpn_2.md.2b6166b8.js} (97%) rename assets/{academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.lean.js => academic_physics_ipho-formulas-jpn_2.md.2b6166b8.lean.js} (97%) rename assets/{academic_physics_ipho-formulas-jpn_3.md.417386cd.js => academic_physics_ipho-formulas-jpn_3.md.24fb7253.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_3.md.417386cd.lean.js => academic_physics_ipho-formulas-jpn_3.md.24fb7253.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.js => academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.lean.js => academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_5.md.d14a555b.js => academic_physics_ipho-formulas-jpn_5.md.db50d685.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_5.md.d14a555b.lean.js => academic_physics_ipho-formulas-jpn_5.md.db50d685.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_6.md.02f25efb.js => academic_physics_ipho-formulas-jpn_6.md.2e655de1.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_6.md.02f25efb.lean.js => academic_physics_ipho-formulas-jpn_6.md.2e655de1.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_7.md.95da3e92.js => academic_physics_ipho-formulas-jpn_7.md.7be7b512.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_7.md.95da3e92.lean.js => academic_physics_ipho-formulas-jpn_7.md.7be7b512.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_8.md.544b0b43.js => academic_physics_ipho-formulas-jpn_8.md.02aeb40e.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_8.md.544b0b43.lean.js => academic_physics_ipho-formulas-jpn_8.md.02aeb40e.lean.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_9.md.6637313d.js => academic_physics_ipho-formulas-jpn_9.md.fc474d28.js} (99%) rename assets/{academic_physics_ipho-formulas-jpn_9.md.6637313d.lean.js => academic_physics_ipho-formulas-jpn_9.md.fc474d28.lean.js} (99%) rename assets/{academic_vocabulary_2023_02_2023-02-27.md.07288c48.js => academic_vocabulary_2023_02_2023-02-27.md.50b28361.js} (98%) rename assets/{academic_vocabulary_2023_02_2023-02-27.md.07288c48.lean.js => academic_vocabulary_2023_02_2023-02-27.md.50b28361.lean.js} (89%) rename assets/{academic_vocabulary_index.md.d46af4f9.js => academic_vocabulary_index.md.b0a1288c.js} (90%) rename assets/{academic_vocabulary_index.md.d46af4f9.lean.js => academic_vocabulary_index.md.b0a1288c.lean.js} (90%) rename assets/{application_markdown-it-katex_how-to-use.md.ae049342.js => application_markdown-it-katex_how-to-use.md.b720171d.js} (97%) rename assets/{application_markdown-it-katex_how-to-use.md.ae049342.lean.js => application_markdown-it-katex_how-to-use.md.b720171d.lean.js} (98%) rename assets/{application_markdown-it-katex_support-function.md.aeaaab84.js => application_markdown-it-katex_support-function.md.1d1d4129.js} (99%) rename assets/{application_markdown-it-katex_support-function.md.aeaaab84.lean.js => application_markdown-it-katex_support-function.md.1d1d4129.lean.js} (99%) rename assets/{application_markdown-it-katex_support-table.md.ff72261b.js => application_markdown-it-katex_support-table.md.646afbf1.js} (99%) rename assets/{application_markdown-it-katex_support-table.md.ff72261b.lean.js => application_markdown-it-katex_support-table.md.646afbf1.lean.js} (99%) rename assets/{application_markdown-it-katex_tips.md.e64f6d16.js => application_markdown-it-katex_tips.md.2d4f69d7.js} (99%) rename assets/{application_markdown-it-katex_tips.md.e64f6d16.lean.js => application_markdown-it-katex_tips.md.2d4f69d7.lean.js} (92%) rename assets/{development_aws_README.md.6ebeba49.js => development_aws_README.md.ea2d6a9b.js} (96%) rename assets/{development_aws_README.md.6ebeba49.lean.js => development_aws_README.md.ea2d6a9b.lean.js} (70%) rename assets/{development_aws_acknowledgement.md.d9cfe93b.js => development_aws_acknowledgement.md.2c27a486.js} (97%) rename assets/{development_aws_acknowledgement.md.d9cfe93b.lean.js => development_aws_acknowledgement.md.2c27a486.lean.js} (86%) rename assets/{development_aws_appendix.md.79df306e.js => development_aws_appendix.md.7272d46f.js} (99%) rename assets/{development_aws_appendix.md.79df306e.lean.js => development_aws_appendix.md.7272d46f.lean.js} (89%) rename assets/{development_aws_assignments.md.3f42c3cd.js => development_aws_assignments.md.27524939.js} (90%) rename assets/{development_aws_assignments.md.3f42c3cd.lean.js => development_aws_assignments.md.27524939.lean.js} (90%) rename assets/{development_aws_author.md.09627572.js => development_aws_author.md.a50d2f4c.js} (95%) rename assets/{development_aws_author.md.09627572.lean.js => development_aws_author.md.a50d2f4c.lean.js} (95%) rename assets/{development_aws_aws-batch.md.83297645.js => development_aws_aws-batch.md.2225dbd4.js} (99%) rename assets/{development_aws_aws-batch.md.83297645.lean.js => development_aws_aws-batch.md.2225dbd4.lean.js} (91%) rename assets/{development_aws_aws-get-started.md.df995644.js => development_aws_aws-get-started.md.305acfd1.js} (99%) rename assets/{development_aws_aws-get-started.md.df995644.lean.js => development_aws_aws-get-started.md.305acfd1.lean.js} (89%) rename assets/{development_aws_closing.md.b2d7e915.js => development_aws_closing.md.73173906.js} (89%) rename assets/{development_aws_closing.md.b2d7e915.lean.js => development_aws_closing.md.73173906.lean.js} (89%) rename assets/{development_aws_cloud.md.34efce50.js => development_aws_cloud.md.d6418ba0.js} (99%) rename assets/{development_aws_cloud.md.34efce50.lean.js => development_aws_cloud.md.d6418ba0.lean.js} (87%) rename assets/{development_aws_docker-system.md.209ef2e6.js => development_aws_docker-system.md.b4b42fe0.js} (99%) rename assets/{development_aws_docker-system.md.209ef2e6.lean.js => development_aws_docker-system.md.b4b42fe0.lean.js} (88%) rename assets/{development_aws_handson-bashoutter.md.4863f098.js => development_aws_handson-bashoutter.md.91af8955.js} (99%) rename assets/{development_aws_handson-bashoutter.md.4863f098.lean.js => development_aws_handson-bashoutter.md.91af8955.lean.js} (89%) rename assets/{development_aws_handson-ec2.md.64c416d6.js => development_aws_handson-ec2.md.d527a33c.js} (99%) rename assets/{development_aws_handson-ec2.md.64c416d6.lean.js => development_aws_handson-ec2.md.d527a33c.lean.js} (90%) rename assets/{development_aws_handson-jupyter.md.c0838dc1.js => development_aws_handson-jupyter.md.887c5eff.js} (99%) rename assets/{development_aws_handson-jupyter.md.c0838dc1.lean.js => development_aws_handson-jupyter.md.887c5eff.lean.js} (90%) rename assets/{development_aws_handson-qabot.md.2f7eac1b.js => development_aws_handson-qabot.md.b244b90e.js} (99%) rename assets/{development_aws_handson-qabot.md.2f7eac1b.lean.js => development_aws_handson-qabot.md.b244b90e.lean.js} (90%) rename assets/{development_aws_handson-serverless.md.38596c90.js => development_aws_handson-serverless.md.a47ced9b.js} (99%) rename assets/{development_aws_handson-serverless.md.38596c90.lean.js => development_aws_handson-serverless.md.a47ced9b.lean.js} (90%) rename assets/{development_aws_introduction.md.222bee89.js => development_aws_introduction.md.a28e3444.js} (99%) rename assets/{development_aws_introduction.md.222bee89.lean.js => development_aws_introduction.md.a28e3444.lean.js} (87%) rename assets/{development_aws_license.md.48a0d402.js => development_aws_license.md.3681aae7.js} (94%) rename assets/{development_aws_license.md.48a0d402.lean.js => development_aws_license.md.3681aae7.lean.js} (94%) rename assets/{development_aws_main.md.dd2f6ba8.js => development_aws_main.md.cc9bca88.js} (99%) rename assets/{development_aws_main.md.dd2f6ba8.lean.js => development_aws_main.md.cc9bca88.lean.js} (96%) rename assets/{development_aws_scientific-computing.md.6e6683eb.js => development_aws_scientific-computing.md.44b26ae4.js} (99%) rename assets/{development_aws_scientific-computing.md.6e6683eb.lean.js => development_aws_scientific-computing.md.44b26ae4.lean.js} (77%) rename assets/{development_aws_serverless.md.7d293d5a.js => development_aws_serverless.md.954dd7a1.js} (99%) rename assets/{development_aws_serverless.md.7d293d5a.lean.js => development_aws_serverless.md.954dd7a1.lean.js} (88%) rename assets/{development_aws_webserver.md.ab41d017.js => development_aws_webserver.md.a8b089c4.js} (99%) rename assets/{development_aws_webserver.md.ab41d017.lean.js => development_aws_webserver.md.a8b089c4.lean.js} (87%) create mode 100644 assets/development_file-naming-convention.md.404e82de.js create mode 100644 assets/development_file-naming-convention.md.404e82de.lean.js create mode 100644 assets/development_rclone-for-r2.md.e7f3a9b7.js create mode 100644 assets/development_rclone-for-r2.md.e7f3a9b7.lean.js rename assets/{getting-started.md.bd1ab99e.js => getting-started.md.ad0edf0e.js} (99%) rename assets/{getting-started.md.bd1ab99e.lean.js => getting-started.md.ad0edf0e.lean.js} (89%) rename assets/{index.md.6f6f33ad.js => index.md.4559d5f1.js} (83%) rename assets/{index.md.6f6f33ad.lean.js => index.md.4559d5f1.lean.js} (83%) rename assets/{javascript_notes_1_1-1.md.10b5b6ea.js => javascript_notes_1_1-1.md.7c957dab.js} (99%) rename assets/{javascript_notes_1_1-1.md.10b5b6ea.lean.js => javascript_notes_1_1-1.md.7c957dab.lean.js} (99%) rename assets/{javascript_notes_1_1-2.md.5a4dd0dd.js => javascript_notes_1_1-2.md.46f1230f.js} (84%) rename assets/{javascript_notes_1_1-2.md.5a4dd0dd.lean.js => javascript_notes_1_1-2.md.46f1230f.lean.js} (84%) rename assets/{roadmap.md.7ba79349.js => roadmap.md.a42ebae3.js} (98%) rename assets/{roadmap.md.7ba79349.lean.js => roadmap.md.a42ebae3.lean.js} (83%) rename assets/{save_reading_index.md.78fc5e98.js => save_reading_index.md.983fe09d.js} (90%) rename assets/{save_reading_index.md.78fc5e98.lean.js => save_reading_index.md.983fe09d.lean.js} (90%) rename assets/{save_reading_outliers_1.md.daf781a1.js => save_reading_outliers_1.md.e3cd2c6d.js} (99%) rename assets/{save_reading_outliers_1.md.daf781a1.lean.js => save_reading_outliers_1.md.e3cd2c6d.lean.js} (91%) rename assets/{save_reading_outliers_2.md.8febfe5f.js => save_reading_outliers_2.md.ad1783d7.js} (99%) rename assets/{save_reading_outliers_2.md.8febfe5f.lean.js => save_reading_outliers_2.md.ad1783d7.lean.js} (90%) rename assets/{save_reading_outliers_3.md.6e397ae9.js => save_reading_outliers_3.md.e6fcee78.js} (99%) rename assets/{save_reading_outliers_3.md.6e397ae9.lean.js => save_reading_outliers_3.md.e6fcee78.lean.js} (90%) rename assets/{save_reading_outliers_4.md.a4a3d333.js => save_reading_outliers_4.md.62591abe.js} (99%) rename assets/{save_reading_outliers_4.md.a4a3d333.lean.js => save_reading_outliers_4.md.62591abe.lean.js} (90%) rename assets/{style.0b53ab5c.css => style.250ef51a.css} (62%) create mode 100644 development/file-naming-convention.html create mode 100644 development/rclone-for-r2.html diff --git a/404.html b/404.html index 87fb2465..9358394d 100644 --- a/404.html +++ b/404.html @@ -5,7 +5,7 @@ 404 | Toshiki's Note - + @@ -30,8 +30,8 @@
Toshiki's Note

Appearance

404

PAGE NOT FOUND

But if you don't change your direction, and if you keep looking, you may end up where you are heading.
Take me home
- + \ No newline at end of file diff --git a/academic/chemistry/index.html b/academic/chemistry/index.html index 60391ce9..edc1b2c2 100644 --- a/academic/chemistry/index.html +++ b/academic/chemistry/index.html @@ -5,12 +5,12 @@ Welcome to Chemistry | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Welcome to Chemistry ​

- +
Toshiki's Note

Appearance

TOC

Welcome to Chemistry ​

+ \ No newline at end of file diff --git a/academic/chemistry/notes/12-5.html b/academic/chemistry/notes/12-5.html index b643da73..02206be2 100644 --- a/academic/chemistry/notes/12-5.html +++ b/academic/chemistry/notes/12-5.html @@ -5,12 +5,12 @@ 12-5: Reaction Mechanism | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

12-5: Reaction Mechanism ​

12-5-1: Learning Objectives ​

Learning Objectives

One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.

In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:

2C8H18(l)+25O2( g)⟶16CO2( g)+18H2O(g)2 \mathrm{C}_{8} \mathrm{H}_{18}(\mathrm{l})+25 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 16 \mathrm{CO}_{2}(\mathrm{~g})+18 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})

For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.

The overall sequence of elementary reactions is the mechanism of the reaction.

12-5-2: Molecularity and the Rate-Determining Step ​

To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.

2CO+2NO2→2CO2+N2O4\mathrm{2CO + 2NO_2 \rightarrow 2CO_2 + N_2O_4}

From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of NO2\mathrm{NO}_{2} with a molecule of CO\mathrm{CO} that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:

 rate =k[NO2]2\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2}

The fact that the reaction is second order in [NO2]\left[\mathrm{NO}_{2}\right] and independent of [CO][\mathrm{CO}] tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be

 rate =k[NO2][CO].\text { rate }=k\left[\mathrm{NO}_{2}\right][\mathrm{CO}] .

The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:

15-2-1: Two-Step Mechanism ​

StepsReactionReaction Type
step 1NO2+NO2⟶ slow NO3+NO\mathrm{NO}_{2}+\mathrm{NO}_{2} \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}_{3}+\mathrm{NO}elementary reaction
step 2NO3+CO→NO2+CO2‾\underline{\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}elementary reaction
sumNO2+CO→NO+CO2\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{NO}+\mathrm{CO}_{2}overall reaction

According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The NO3\mathrm{NO_3} molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.

The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.

12-5-3: Using Molecularity to Describe a Rate Law ​

The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)

About the image

The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)

Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( A→\mathrm{A} \rightarrow products) is

 rate =k[A].\text { rate }=k[A] .

For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form A+B→\mathrm{A}+\mathrm{B} \rightarrow products, the general rate law is

 rate =k[A][B].\text { rate }=k[A][B] .

Elementary ReactionMolecularityRateReaction Order
A→ products \mathrm{A} \rightarrow \text { products }Unimolecular rate =k[ A]\text { rate }=k[\mathrm{~A}]first
2 A→ products 2\mathrm{~A} \rightarrow \text { products }Bimolecular rate =k[ A]2\text { rate }=k[\mathrm{~A}]^2second
A+B→ products \mathrm{A}+\mathrm{B} \rightarrow \text { products }Bimolecular rate =k[ A][B]\text { rate }=k[\mathrm{~A}][\mathrm{B}]second
2 A+B→ products 2 \mathrm{~A}+\mathrm{B} \rightarrow \text { products }Termolecular rate =k[ A]2[ B]\text { rate }=k[\mathrm{~A}]^2[\mathrm{~B}]third
A+B+C→ products \mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow \text { products }Termolecular rate =k[ A][B][C]\text { rate }=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}]third

For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).

12-5-4: Identifying the Rate-Determining Step ​

Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.

Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.

Rate laws for each elementary reaction in our example as well as for the overall reaction

StepsReactionRate
step 1NO2+NO2→k1NO3+NO\mathrm{NO}_2+\mathrm{NO}_2 \stackrel{\mathrm{k}_1}{\rightarrow} \mathrm{NO}_3+\mathrm{NO} rate =k1[NO2]2( predicted) \text { rate }=k_1\left[\mathrm{NO}_2\right]^2(\text { predicted) }
step 2NO3+CO→k2NO2+CO2‾\underline{\mathrm{NO}_3+\mathrm{CO} \stackrel{k_2}{\rightarrow} \mathrm{NO}_2+\mathrm{CO}_2} rate =k2[NO3][CO]( predicted )\text { rate }=k_2\left[\mathrm{NO}_3\right][\mathrm{CO}](\text { predicted })
step 3NO2+CO→kNO+CO2\mathrm{NO}_2+\mathrm{CO} \stackrel{k}{\rightarrow} \mathrm{NO}+\mathrm{CO}_2 rate =k[NO2]2( observed) \text { rate }=k\left[\mathrm{NO}_2\right]^2(\text { observed) }

The experimentally determined rate law for the reaction of NO2\mathrm{NO}_{2} with COC O is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so kk for the overall reaction must equal k1k_{1}. That is, NO3\mathrm{NO}_{3} is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.

Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.

12-5-4-1: Example-A Reaction with an Intermediate ​

In an alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate =k[NO2]2t=\mathrm{k[{NO}_{2}]^{2}t})

Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law

Strategy ​

  • Determine the rate law for each elementary reaction in the reaction.

  • Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.

Solution ​

View solution

A The rate law for step 1 is rate =k1[NO2]2=k_{1}\left[\mathrm{NO}_{2}\right]^{2}; for step 2 , it is rate =k2[ N2O4][CO]=k_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{4}\right][\mathrm{CO}].

B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate == k1[NO2]2k_{1}\left[\mathrm{NO}_{2}\right]^{2}. This is the same as the experimentally determined rate law. Hence this mechanism, with N2O4\mathrm{N}_{2} \mathrm{O}_{4} as an intermediate, and the one described previously, with NO3\mathrm{NO}_{3} as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, NO3\mathrm{NO}_{3} and N2O4\mathrm{N}_{2} \mathrm{O}_{4}, directly.

- +
Toshiki's Note

Appearance

TOC

12-5: Reaction Mechanism ​

12-5-1: Learning Objectives ​

Learning Objectives

One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.

In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:

2C8H18(l)+25O2( g)⟶16CO2( g)+18H2O(g)2 \mathrm{C}_{8} \mathrm{H}_{18}(\mathrm{l})+25 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 16 \mathrm{CO}_{2}(\mathrm{~g})+18 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})

For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.

The overall sequence of elementary reactions is the mechanism of the reaction.

12-5-2: Molecularity and the Rate-Determining Step ​

To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.

2CO+2NO2→2CO2+N2O4\mathrm{2CO + 2NO_2 \rightarrow 2CO_2 + N_2O_4}

From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of NO2\mathrm{NO}_{2} with a molecule of CO\mathrm{CO} that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:

 rate =k[NO2]2\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2}

The fact that the reaction is second order in [NO2]\left[\mathrm{NO}_{2}\right] and independent of [CO][\mathrm{CO}] tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be

 rate =k[NO2][CO].\text { rate }=k\left[\mathrm{NO}_{2}\right][\mathrm{CO}] .

The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:

15-2-1: Two-Step Mechanism ​

StepsReactionReaction Type
step 1NO2+NO2⟶ slow NO3+NO\mathrm{NO}_{2}+\mathrm{NO}_{2} \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}_{3}+\mathrm{NO}elementary reaction
step 2NO3+CO→NO2+CO2‾\underline{\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}elementary reaction
sumNO2+CO→NO+CO2\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{NO}+\mathrm{CO}_{2}overall reaction

According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The NO3\mathrm{NO_3} molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.

The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.

12-5-3: Using Molecularity to Describe a Rate Law ​

The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)

About the image

The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)

Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( A→\mathrm{A} \rightarrow products) is

 rate =k[A].\text { rate }=k[A] .

For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form A+B→\mathrm{A}+\mathrm{B} \rightarrow products, the general rate law is

 rate =k[A][B].\text { rate }=k[A][B] .

Elementary ReactionMolecularityRateReaction Order
A→ products \mathrm{A} \rightarrow \text { products }Unimolecular rate =k[ A]\text { rate }=k[\mathrm{~A}]first
2 A→ products 2\mathrm{~A} \rightarrow \text { products }Bimolecular rate =k[ A]2\text { rate }=k[\mathrm{~A}]^2second
A+B→ products \mathrm{A}+\mathrm{B} \rightarrow \text { products }Bimolecular rate =k[ A][B]\text { rate }=k[\mathrm{~A}][\mathrm{B}]second
2 A+B→ products 2 \mathrm{~A}+\mathrm{B} \rightarrow \text { products }Termolecular rate =k[ A]2[ B]\text { rate }=k[\mathrm{~A}]^2[\mathrm{~B}]third
A+B+C→ products \mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow \text { products }Termolecular rate =k[ A][B][C]\text { rate }=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}]third

For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).

12-5-4: Identifying the Rate-Determining Step ​

Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.

Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.

Rate laws for each elementary reaction in our example as well as for the overall reaction

StepsReactionRate
step 1NO2+NO2→k1NO3+NO\mathrm{NO}_2+\mathrm{NO}_2 \stackrel{\mathrm{k}_1}{\rightarrow} \mathrm{NO}_3+\mathrm{NO} rate =k1[NO2]2( predicted) \text { rate }=k_1\left[\mathrm{NO}_2\right]^2(\text { predicted) }
step 2NO3+CO→k2NO2+CO2‾\underline{\mathrm{NO}_3+\mathrm{CO} \stackrel{k_2}{\rightarrow} \mathrm{NO}_2+\mathrm{CO}_2} rate =k2[NO3][CO]( predicted )\text { rate }=k_2\left[\mathrm{NO}_3\right][\mathrm{CO}](\text { predicted })
step 3NO2+CO→kNO+CO2\mathrm{NO}_2+\mathrm{CO} \stackrel{k}{\rightarrow} \mathrm{NO}+\mathrm{CO}_2 rate =k[NO2]2( observed) \text { rate }=k\left[\mathrm{NO}_2\right]^2(\text { observed) }

The experimentally determined rate law for the reaction of NO2\mathrm{NO}_{2} with COC O is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so kk for the overall reaction must equal k1k_{1}. That is, NO3\mathrm{NO}_{3} is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.

Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.

12-5-4-1: Example-A Reaction with an Intermediate ​

In an alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

alternative mechanism for the reaction of NO2\mathrm{NO}_{2} with CO\mathrm{CO} with N2O4\mathrm{N}_{2} \mathrm{O}_{4} appearing as an intermediate.

Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate =k[NO2]2t=\mathrm{k[{NO}_{2}]^{2}t})

Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law

Strategy ​

  • Determine the rate law for each elementary reaction in the reaction.

  • Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.

Solution ​

View solution

A The rate law for step 1 is rate =k1[NO2]2=k_{1}\left[\mathrm{NO}_{2}\right]^{2}; for step 2 , it is rate =k2[ N2O4][CO]=k_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{4}\right][\mathrm{CO}].

B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate == k1[NO2]2k_{1}\left[\mathrm{NO}_{2}\right]^{2}. This is the same as the experimentally determined rate law. Hence this mechanism, with N2O4\mathrm{N}_{2} \mathrm{O}_{4} as an intermediate, and the one described previously, with NO3\mathrm{NO}_{3} as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, NO3\mathrm{NO}_{3} and N2O4\mathrm{N}_{2} \mathrm{O}_{4}, directly.

+ \ No newline at end of file diff --git a/academic/chemistry/problems/02-20.html b/academic/chemistry/problems/02-20.html index 78e8cd5a..45c149f0 100644 --- a/academic/chemistry/problems/02-20.html +++ b/academic/chemistry/problems/02-20.html @@ -5,12 +5,12 @@ Presentation problem: 02-20 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Presentation problem: 02-20 ​

Question ​

At 500 K500 \mathrm{~K} in the presence of a copper surface, ethanol decomposes according to the equation

C2H5OH(g)⟶CH3CHO(g)+H2(g)\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{g}) \longrightarrow \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2(g)

The pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} was measured as a function of time and the following data were obtained:

Time (s)PC2H5OH (torr) P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }
0250.
100.237
200.224
300.211
400.198
500.185

Since the pressure of a gas is directly proportional to the concentration of gas, we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} after 900.s900 . \mathrm{s} from the start of the reaction. (Hint: To determine the order of the reaction with respect to C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}, compare how the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} decreases with each time listing.)

Solution ​

solution graph
◎

Due to the fact that the graph of p[O2]\mathrm{p}\left[\mathrm{O}_2\right] over time is showing R2\mathrm{R}^2 value of 1 we know we have a zero-order reaction. Therefore:

k=p0[C2H5OH]−p[C2H5OH]t=250 torr −185 torr 500 s=0.13 torr s −1k=0.13 torr s s−1×1 atm760 torr =1.71×10−4 atm s−1 rate =kp(C2H5OH)=−kt+p0(C2H5OH)p(C2H5OH)=−(1.71×10−4 atm s−1)×900 s+0.33 atmp(C2H5OH)=0.176 atm s−1\begin{aligned} & \mathrm{k}=\frac{\mathrm{p}_0\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]-\mathrm{p}\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]}{\mathrm{t}}=\frac{250 \text { torr }-185 \text { torr }}{500 \mathrm{~s}}=0.13 \text { torr s }^{-1} \\ & \mathrm{k}=0.13 \text { torr s } \mathrm{s}^{-1} \times \frac{1 \mathrm{~atm}}{760 \text { torr }}=1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1} \\ & \text { rate }=\mathbf{k} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\mathrm{kt}+\mathrm{p}_0\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right) \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\left(1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1}\right) \times 900 \mathrm{~s}+0.33 \mathrm{~atm} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ & \end{aligned}

- +
Toshiki's Note

Appearance

TOC

Presentation problem: 02-20 ​

Question ​

At 500 K500 \mathrm{~K} in the presence of a copper surface, ethanol decomposes according to the equation

C2H5OH(g)⟶CH3CHO(g)+H2(g)\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{g}) \longrightarrow \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2(g)

The pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} was measured as a function of time and the following data were obtained:

Time (s)PC2H5OH (torr) P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }
0250.
100.237
200.224
300.211
400.198
500.185

Since the pressure of a gas is directly proportional to the concentration of gas, we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} after 900.s900 . \mathrm{s} from the start of the reaction. (Hint: To determine the order of the reaction with respect to C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}, compare how the pressure of C2H5OH\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} decreases with each time listing.)

Solution ​

solution graph
◎

Due to the fact that the graph of p[O2]\mathrm{p}\left[\mathrm{O}_2\right] over time is showing R2\mathrm{R}^2 value of 1 we know we have a zero-order reaction. Therefore:

k=p0[C2H5OH]−p[C2H5OH]t=250 torr −185 torr 500 s=0.13 torr s −1k=0.13 torr s s−1×1 atm760 torr =1.71×10−4 atm s−1 rate =kp(C2H5OH)=−kt+p0(C2H5OH)p(C2H5OH)=−(1.71×10−4 atm s−1)×900 s+0.33 atmp(C2H5OH)=0.176 atm s−1\begin{aligned} & \mathrm{k}=\frac{\mathrm{p}_0\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]-\mathrm{p}\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]}{\mathrm{t}}=\frac{250 \text { torr }-185 \text { torr }}{500 \mathrm{~s}}=0.13 \text { torr s }^{-1} \\ & \mathrm{k}=0.13 \text { torr s } \mathrm{s}^{-1} \times \frac{1 \mathrm{~atm}}{760 \text { torr }}=1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1} \\ & \text { rate }=\mathbf{k} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\mathrm{kt}+\mathrm{p}_0\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right) \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=-\left(1.71 \times 10^{-4} \mathrm{~atm} \mathrm{~s}^{-1}\right) \times 900 \mathrm{~s}+0.33 \mathrm{~atm} \\ & \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ & \end{aligned}

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-1.html b/academic/chemistry/problems/03-02-1.html index 53fa69ee..7d5921a2 100644 --- a/academic/chemistry/problems/03-02-1.html +++ b/academic/chemistry/problems/03-02-1.html @@ -5,12 +5,12 @@ Problem: 03-02-1 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Problem: 03-02-1 ​

Question ​

2 N2O5(g)→4NO2(g)+O2(g)2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g)

The decomposition of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is represented by the equation above. A sample of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is monitored as it decomposes, and the concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 as a function of time is recorded. The results are shown in the table below.

Time (s)[N2O5]\mathrm{[N_2O_5]}
01.000
25.00.801
50.00.642
75.00.515

Calculate the average rate of the reaction between 50.050.0 and 75.075.0 seconds.

Solution ​

a A+b B→cC+dDa \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D}

the rate of reaction is defined as

 rate =−1aΔ[A]Δt=−1bΔ[B]Δt=+1cΔ[C]Δt=+1dΔ[D]Δt\color{cyan}\text { rate }=-\frac{1}{a} \frac{\Delta[\mathrm{A}]}{\Delta t}=-\frac{1}{b} \frac{\Delta[\mathrm{B}]}{\Delta t}=+\frac{1}{c} \frac{\Delta[\mathrm{C}]}{\Delta t}=+\frac{1}{d} \frac{\Delta[\mathrm{D}]}{\Delta t}

Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( aa, b,cb, c, or dd ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was N2O5\mathrm{N}_2 \mathrm{O}_5. With that in mind, let's write the reaction rate in terms of the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 :

 rate =−12Δ[N2O5]Δt\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t}

Since the coefficient for N2O5\mathrm{N}_2 \mathrm{O}_5 in the balanced equation is 2 , we divided the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 by 2 . Additionally, since N2O5\mathrm{N}_2 \mathrm{O}_5 is being consumed in the reaction, we included a negative sign in front of the expression.

Now, let's plug in the information from the table to calculate the average reaction rate between 50.050.0 and 75.075.0 seconds:

 rate =−12(0.515M−0.642M)(75.0 s−50.0 s)=2.54×10−3M s−1\begin{aligned} \text { rate } & =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ & =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned}

So, the average rate of the reaction between 50.050.0 and 75.075.0 seconds is 2.54×10−3Ms−1\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}.

- +
Toshiki's Note

Appearance

TOC

Problem: 03-02-1 ​

Question ​

2 N2O5(g)→4NO2(g)+O2(g)2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g)

The decomposition of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is represented by the equation above. A sample of N2O5(g)\mathrm{N}_2 \mathrm{O}_5(g) is monitored as it decomposes, and the concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 as a function of time is recorded. The results are shown in the table below.

Time (s)[N2O5]\mathrm{[N_2O_5]}
01.000
25.00.801
50.00.642
75.00.515

Calculate the average rate of the reaction between 50.050.0 and 75.075.0 seconds.

Solution ​

a A+b B→cC+dDa \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D}

the rate of reaction is defined as

 rate =−1aΔ[A]Δt=−1bΔ[B]Δt=+1cΔ[C]Δt=+1dΔ[D]Δt\color{cyan}\text { rate }=-\frac{1}{a} \frac{\Delta[\mathrm{A}]}{\Delta t}=-\frac{1}{b} \frac{\Delta[\mathrm{B}]}{\Delta t}=+\frac{1}{c} \frac{\Delta[\mathrm{C}]}{\Delta t}=+\frac{1}{d} \frac{\Delta[\mathrm{D}]}{\Delta t}

Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( aa, b,cb, c, or dd ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was N2O5\mathrm{N}_2 \mathrm{O}_5. With that in mind, let's write the reaction rate in terms of the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 :

 rate =−12Δ[N2O5]Δt\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t}

Since the coefficient for N2O5\mathrm{N}_2 \mathrm{O}_5 in the balanced equation is 2 , we divided the rate of change in concentration of N2O5\mathrm{N}_2 \mathrm{O}_5 by 2 . Additionally, since N2O5\mathrm{N}_2 \mathrm{O}_5 is being consumed in the reaction, we included a negative sign in front of the expression.

Now, let's plug in the information from the table to calculate the average reaction rate between 50.050.0 and 75.075.0 seconds:

 rate =−12(0.515M−0.642M)(75.0 s−50.0 s)=2.54×10−3M s−1\begin{aligned} \text { rate } & =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ & =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned}

So, the average rate of the reaction between 50.050.0 and 75.075.0 seconds is 2.54×10−3Ms−1\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}.

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-2.html b/academic/chemistry/problems/03-02-2.html index 0bf013b9..d58665d8 100644 --- a/academic/chemistry/problems/03-02-2.html +++ b/academic/chemistry/problems/03-02-2.html @@ -5,12 +5,12 @@ Problem: 03-02-2 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Problem: 03-02-2 ​

Question ​

The rate law for a particular reaction is rate =k[XY]2=k[\mathrm{XY}]^2. In an experiment, the initial rate of the reaction is determined to be 0.16 mol/(L⋅s)0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s}) when the initial concentration of XY\mathrm{XY} is 0.40 mol/L0.40 \mathrm{~mol} / \mathrm{L}.

  • What is the value of the rate constant, kk, for the reaction?
    • 0.10 L/(mol⋅s)0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 0.40 L/(mol⋅s)0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 2.5 L/(mol⋅s)2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

Solution ​

lo tind the value of the rate constant for the reaction, let's first solve the rate law for kk :

k= rate [XY]2k=\frac{\text { rate }}{[\mathrm{XY}]^2}

Next, let's plug in the initial rate and concentration given in the text:

k=0.16 mol/(L⋅s)(0.40 mol/L)2=0.16 mol/(L⋅s)0.16 mol2/L2=1.0 L/(mol⋅s)\begin{aligned} k & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ & =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned}

So, the value of the rate constant for the reaction is 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

- +
Toshiki's Note

Appearance

TOC

Problem: 03-02-2 ​

Question ​

The rate law for a particular reaction is rate =k[XY]2=k[\mathrm{XY}]^2. In an experiment, the initial rate of the reaction is determined to be 0.16 mol/(L⋅s)0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s}) when the initial concentration of XY\mathrm{XY} is 0.40 mol/L0.40 \mathrm{~mol} / \mathrm{L}.

  • What is the value of the rate constant, kk, for the reaction?
    • 0.10 L/(mol⋅s)0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 0.40 L/(mol⋅s)0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})
    • 2.5 L/(mol⋅s)2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

Solution ​

lo tind the value of the rate constant for the reaction, let's first solve the rate law for kk :

k= rate [XY]2k=\frac{\text { rate }}{[\mathrm{XY}]^2}

Next, let's plug in the initial rate and concentration given in the text:

k=0.16 mol/(L⋅s)(0.40 mol/L)2=0.16 mol/(L⋅s)0.16 mol2/L2=1.0 L/(mol⋅s)\begin{aligned} k & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ & =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ & =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned}

So, the value of the rate constant for the reaction is 1.0 L/(mol⋅s)1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})

+ \ No newline at end of file diff --git a/academic/chemistry/problems/03-02-3.html b/academic/chemistry/problems/03-02-3.html index 54d28819..6dbabaae 100644 --- a/academic/chemistry/problems/03-02-3.html +++ b/academic/chemistry/problems/03-02-3.html @@ -5,12 +5,12 @@ Problem 03-02-3 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Problem 03-02-3 ​

Question ​

2X(g)+2Y(g)→Q(g)+2R(g)2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g)

The reaction represented above is found to be second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}.

What happens to the rate of the reaction when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled?

  • It increases by a factor of 4.
  • It decreases by a factor of 2.
  • It decreases by a factor of 4.
  • It does not change.

Solution ​

To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}, so the rate law is rate =k[X]2[Y]=k[\mathrm{X}]^2[\mathrm{Y}]

Now, let's think about how the rate changes when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled. Since [X][\mathrm{X}] is raised to the second power in the rate law, halving [X][\mathrm{X}] decreases the reaction rate by a factor of 4 . Similarly, since [Y][\mathrm{Y}] is raised to the first power, doubling [Y][\mathrm{Y}] increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled, the rate of the reaction decreases by a factor of 2 .

- +
Toshiki's Note

Appearance

TOC

Problem 03-02-3 ​

Question ​

2X(g)+2Y(g)→Q(g)+2R(g)2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g)

The reaction represented above is found to be second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}.

What happens to the rate of the reaction when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled?

  • It increases by a factor of 4.
  • It decreases by a factor of 2.
  • It decreases by a factor of 4.
  • It does not change.

Solution ​

To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to X\mathrm{X} and first order with respect to Y\mathrm{Y}, so the rate law is rate =k[X]2[Y]=k[\mathrm{X}]^2[\mathrm{Y}]

Now, let's think about how the rate changes when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled. Since [X][\mathrm{X}] is raised to the second power in the rate law, halving [X][\mathrm{X}] decreases the reaction rate by a factor of 4 . Similarly, since [Y][\mathrm{Y}] is raised to the first power, doubling [Y][\mathrm{Y}] increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when [X][\mathrm{X}] is halved and [Y][\mathrm{Y}] is doubled, the rate of the reaction decreases by a factor of 2 .

+ \ No newline at end of file diff --git a/academic/literature/index.html b/academic/literature/index.html index 51266066..35ed92d5 100644 --- a/academic/literature/index.html +++ b/academic/literature/index.html @@ -5,12 +5,12 @@ Welcome to Literature | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Welcome to Literature ​

- +
Toshiki's Note

Appearance

TOC

Welcome to Literature ​

+ \ No newline at end of file diff --git a/academic/literature/writing/methods-of-development.html b/academic/literature/writing/methods-of-development.html index d9b150c6..b042d357 100644 --- a/academic/literature/writing/methods-of-development.html +++ b/academic/literature/writing/methods-of-development.html @@ -5,12 +5,12 @@ Patterns of Organization and Methods of Development | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Patterns of Organization and Methods of Development ​

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

  • Now take the pie out of the oven and let it cool on the stovetop.
  • Mix the dry ingredients with the liquid ingredients.
  • Set the pie crust aside while you make the filling.

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

  • Most important information first (consider what you want readers to focus on first)
  • Chronological order (the order in time that events take place)
  • Compare and contrast (ideas are organized together because of their relationship to each other)

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

  • Facing fears
  • Safety in sports
  • Community policing
  • Educating prisoners
  • Sex education

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

  • Describe the problem, followed by the solution.
  • Propose the solution first and then describe the problems that motivated it.
  • Or a problem may be followed by several solutions, one of which is selected as the best.

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

  • This chapter was adapted from “Patterns of Organization and Methods of Development” in The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a CC BY-NC 4.0 Licence. Adapted by Allison Kilgannon.

  • “Peach and lavender pie” by Heather Joan is licensed under a CC BY-NC-ND 2.0 Licence.

  • “General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a CC BY-NC 4.0 Licence.

  • Kilgannon, Allison. “Patterns of Organization and Methods of Development.” Opentextbc.ca, 20 Aug. 2021, opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development.

- +
Toshiki's Note

Appearance

TOC

Patterns of Organization and Methods of Development ​

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

  • Now take the pie out of the oven and let it cool on the stovetop.
  • Mix the dry ingredients with the liquid ingredients.
  • Set the pie crust aside while you make the filling.

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

  • Most important information first (consider what you want readers to focus on first)
  • Chronological order (the order in time that events take place)
  • Compare and contrast (ideas are organized together because of their relationship to each other)

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

  • Facing fears
  • Safety in sports
  • Community policing
  • Educating prisoners
  • Sex education

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

  • Describe the problem, followed by the solution.
  • Propose the solution first and then describe the problems that motivated it.
  • Or a problem may be followed by several solutions, one of which is selected as the best.

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

  • This chapter was adapted from “Patterns of Organization and Methods of Development” in The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a CC BY-NC 4.0 Licence. Adapted by Allison Kilgannon.

  • “Peach and lavender pie” by Heather Joan is licensed under a CC BY-NC-ND 2.0 Licence.

  • “General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a CC BY-NC 4.0 Licence.

  • Kilgannon, Allison. “Patterns of Organization and Methods of Development.” Opentextbc.ca, 20 Aug. 2021, opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development.

+ \ No newline at end of file diff --git a/academic/physics/index.html b/academic/physics/index.html index 87489c31..43b3afdd 100644 --- a/academic/physics/index.html +++ b/academic/physics/index.html @@ -5,12 +5,12 @@ Welcome to Physics | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Welcome to Physics ​

- +
Toshiki's Note

Appearance

TOC

Welcome to Physics ​

+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/1.html b/academic/physics/ipho-formulas-jpn/1.html index ed9898e7..abc15089 100644 --- a/academic/physics/ipho-formulas-jpn/1.html +++ b/academic/physics/ipho-formulas-jpn/1.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 1 | Toshiki's Note - + - + @@ -52,9 +52,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​​

1.13: 積分 ​

  1. 積分：微分の公式の左辺と右辺を入れ替えたものと同 じ（逆演算）.例えば,

    ∫xn dx=xn+1/(n+1).\int x^n \mathrm{~d} x=x^{n+1} /(n+1) .

    置換積分の特別な場合 :

    ∫f(ax+b)dx=F(ax+b)/a.\int f(a x+b) \mathrm{d} x=F(a x+b) / a .

1.14: 円錐曲線 ​

  1. 円錐曲線: a11x2+2a12xy+a22y2+a1x+a2y+a0=a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0= 0 で, a11=a22a_{11}=a_{22} ならば円, a11(a11a22−a122)>0a_{11}\left(a_{11} a_{22}-a_{12}^2\right)>0 ならば楕円, ⋯<0\cdots<0 ならば双曲線, a11a22−a122=0a_{11} a_{22}-a_{12}^2=0 ならば放物線. 楕円 : l1+l2=2a,α1=α2l_1+l_2=2 a, \alpha_1=\alpha_2 [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], A=πabA=\pi a b. 双曲線 : l1−l2=2a,α1+α2=0l_1-l_2=2 a, \alpha_1+\alpha_2=0. 放物線 :l+h=: l+h= const., α1=α2\alpha_1=\alpha_2.

1.15: 数值計算 & 台形規則 ​

  1. 数值計算. f(x)=0f(x)=0 の解を求めるニュートン法 :

    xn+1=xn−f(xn)/f′(xn)x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right)

    近似積分の台形規則：

    ∫abf(x)dx≈b−a2n[f(x0)+2{f(x1)+⋯ +f(xn−1)}+f(xn)]\begin{array}{r} \int_a^b f(x) \mathrm{d} x \approx \frac{b-a}{2 n}\left[f\left(x_0\right)+2\left\{f\left(x_1\right)+\cdots\right.\right. \left.\left.+f\left(x_{n-1}\right)\right\}+f\left(x_n\right)\right] \end{array}

1.16: ベクトルの微分 & 積分 ​

  1. ベクトルの微分と積分：成分ごとに微分/積分する．あるいは無限に近い22つのベクトルの差を求めることで 微分す.
- +M1001 80h400000v40h-400000z">​​

1.13: 積分 ​

  1. 積分：微分の公式の左辺と右辺を入れ替えたものと同 じ（逆演算）.例えば,

    ∫xn dx=xn+1/(n+1).\int x^n \mathrm{~d} x=x^{n+1} /(n+1) .

    置換積分の特別な場合 :

    ∫f(ax+b)dx=F(ax+b)/a.\int f(a x+b) \mathrm{d} x=F(a x+b) / a .

1.14: 円錐曲線 ​

  1. 円錐曲線: a11x2+2a12xy+a22y2+a1x+a2y+a0=a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0= 0 で, a11=a22a_{11}=a_{22} ならば円, a11(a11a22−a122)>0a_{11}\left(a_{11} a_{22}-a_{12}^2\right)>0 ならば楕円, ⋯<0\cdots<0 ならば双曲線, a11a22−a122=0a_{11} a_{22}-a_{12}^2=0 ならば放物線. 楕円 : l1+l2=2a,α1=α2l_1+l_2=2 a, \alpha_1=\alpha_2 [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], A=πabA=\pi a b. 双曲線 : l1−l2=2a,α1+α2=0l_1-l_2=2 a, \alpha_1+\alpha_2=0. 放物線 :l+h=: l+h= const., α1=α2\alpha_1=\alpha_2.

1.15: 数值計算 & 台形規則 ​

  1. 数值計算. f(x)=0f(x)=0 の解を求めるニュートン法 :

    xn+1=xn−f(xn)/f′(xn)x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right)

    近似積分の台形規則：

    ∫abf(x)dx≈b−a2n[f(x0)+2{f(x1)+⋯ +f(xn−1)}+f(xn)]\begin{array}{r} \int_a^b f(x) \mathrm{d} x \approx \frac{b-a}{2 n}\left[f\left(x_0\right)+2\left\{f\left(x_1\right)+\cdots\right.\right. \left.\left.+f\left(x_{n-1}\right)\right\}+f\left(x_n\right)\right] \end{array}

1.16: ベクトルの微分 & 積分 ​

  1. ベクトルの微分と積分：成分ごとに微分/積分する．あるいは無限に近い22つのベクトルの差を求めることで 微分す.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/10.html b/academic/physics/ipho-formulas-jpn/10.html index 2b732758..df82834d 100644 --- a/academic/physics/ipho-formulas-jpn/10.html +++ b/academic/physics/ipho-formulas-jpn/10.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 10 | Toshiki's Note - + - + @@ -51,9 +51,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​,ν=vnS.

10.10: Carnot サイクル ​

  1. Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. S−TS-T 座標を用いることにより η=(T1−T2)/T1\eta=\left(T_1-T_2\right) / T_1 を得る.

10.11: ヒートポンプ ​

  1. ヒートポンプ: Carnot サイクルの逆. η=T1T1−T2\eta=\frac{T_1}{T_1-T_2}.

10.12: エントロピー ​

  1. エントロピー :dS=dQ/T: \mathrm{d} S=\mathrm{d} Q / T.

10.13: 熱力学第一法則 ​

  1. 熱力学第一法則 : d′U=d′A+d′Q\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q

10.14: 熱力学第二法則 ​

  1. 熱力学第二法則 : ΔS≥0\Delta S \geq 0 (また ηreal ≤ηCarnot )\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right).

10.15: 気体のする仕事 ​

  1. 気体のする仕事（ポイント 10 も参照）:

    A=∫p dV, 断熱過程: A=i2Δ(pV)A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V)

10.16: Dalton の法則 ​

  1. Dalton の法則: p=p= ∑pi\sum p_i

    訳者注

    理想気体のみ成立

10.17: 沸騰 ​

  1. 沸騰: 飽和蒸気の圧力 pv=p0.2p_v=p_0 .2 液の界面では pv1+pv2=p0p_{v 1}+p_{v 2}=p_0.

10.18: 熱流 ​

  1. 熱流: P=kSΔT/lP=k S \Delta T / l ( kk は熱伝導率). 直流回路に似て いる (P↔I,ΔT↔V,k↔1/ρ)(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho).

10.19: 熱容量 ​

  1. 熱容量 : Q=∫c(T)dTQ=\int c(T) \mathrm{d} T. 固体では低温で c∝T3c \propto T^3, 高温で c=3NkBc=3 N k_B （Dulong-Petit の法則. ここで NN は結晶中の原子数）

10.20: 表面張力 ​

  1. 表面張力 :

    U=Sσ,F=lσ,p=2σ/RU=S \sigma, F=l \sigma, p=2 \sigma / R

10.21: Stefan-Boltzmann の法則 (灰色体) ​

  1. Stefan-Boltzmann の法則 (灰色体) : P=εσAT4P=\varepsilon \sigma A T^4.

10.22: Wien の変位則 ​

  1. Wien の変位則: νmax⁡=AkBT/h(A≈\nu_{\max }=A k_B T / h(A \approx 2.8), λmax⁡=hc/A′kBT(A′≈5)\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right).
- +M1001 80h400000v40h-400000z">​,ν=vnS.

10.10: Carnot サイクル ​

  1. Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. S−TS-T 座標を用いることにより η=(T1−T2)/T1\eta=\left(T_1-T_2\right) / T_1 を得る.

10.11: ヒートポンプ ​

  1. ヒートポンプ: Carnot サイクルの逆. η=T1T1−T2\eta=\frac{T_1}{T_1-T_2}.

10.12: エントロピー ​

  1. エントロピー :dS=dQ/T: \mathrm{d} S=\mathrm{d} Q / T.

10.13: 熱力学第一法則 ​

  1. 熱力学第一法則 : d′U=d′A+d′Q\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q

10.14: 熱力学第二法則 ​

  1. 熱力学第二法則 : ΔS≥0\Delta S \geq 0 (また ηreal ≤ηCarnot )\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right).

10.15: 気体のする仕事 ​

  1. 気体のする仕事（ポイント 10 も参照）:

    A=∫p dV, 断熱過程: A=i2Δ(pV)A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V)

10.16: Dalton の法則 ​

  1. Dalton の法則: p=p= ∑pi\sum p_i

    訳者注

    理想気体のみ成立

10.17: 沸騰 ​

  1. 沸騰: 飽和蒸気の圧力 pv=p0.2p_v=p_0 .2 液の界面では pv1+pv2=p0p_{v 1}+p_{v 2}=p_0.

10.18: 熱流 ​

  1. 熱流: P=kSΔT/lP=k S \Delta T / l ( kk は熱伝導率). 直流回路に似て いる (P↔I,ΔT↔V,k↔1/ρ)(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho).

10.19: 熱容量 ​

  1. 熱容量 : Q=∫c(T)dTQ=\int c(T) \mathrm{d} T. 固体では低温で c∝T3c \propto T^3, 高温で c=3NkBc=3 N k_B （Dulong-Petit の法則. ここで NN は結晶中の原子数）

10.20: 表面張力 ​

  1. 表面張力 :

    U=Sσ,F=lσ,p=2σ/RU=S \sigma, F=l \sigma, p=2 \sigma / R

10.21: Stefan-Boltzmann の法則 (灰色体) ​

  1. Stefan-Boltzmann の法則 (灰色体) : P=εσAT4P=\varepsilon \sigma A T^4.

10.22: Wien の変位則 ​

  1. Wien の変位則: νmax⁡=AkBT/h(A≈\nu_{\max }=A k_B T / h(A \approx 2.8), λmax⁡=hc/A′kBT(A′≈5)\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right).
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/11.html b/academic/physics/ipho-formulas-jpn/11.html index 50bd7ba4..ddc8a375 100644 --- a/academic/physics/ipho-formulas-jpn/11.html +++ b/academic/physics/ipho-formulas-jpn/11.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 11 | Toshiki's Note - + - + @@ -42,9 +42,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​) であれば容易に透過する.

11.7: Bohr モデル ​

  1. Bohr モデル : En∝−1/n2E_n \propto-1 / n^2. (古典的に計算される) 円軌道では, 軌道の長さが波長 λ=h/mv\lambda=h / m v の整数倍.

11.8: Compton 効果 ​

  1. Compton 効果: 光子が電子から散乱されると, 光子の Δλ=λC(1−cos⁡θ)\Delta \lambda=\lambda_C(1-\cos \theta)

11.9: 光電効果 ​

  1. 光電効果: W+mvmax⁡2/2=hνW+m v_{\max }^2 / 2=h \nu ( WW は仕事関数 )). I−VI-V グラフ：光電流は阻止電圧 V=−(hν−W)/eV=-(h \nu-W) / e で始まり, 正方向に電圧が大きくなると緩和する.

11.10: Stefan-Boltzmann の法則 ​

  1. Stefan-Boltzmann の法則 : P=σAT4P=\sigma A T^4
- +M1001 80h400000v40h-400000z">​) であれば容易に透過する.

11.7: Bohr モデル ​

  1. Bohr モデル : En∝−1/n2E_n \propto-1 / n^2. (古典的に計算される) 円軌道では, 軌道の長さが波長 λ=h/mv\lambda=h / m v の整数倍.

11.8: Compton 効果 ​

  1. Compton 効果: 光子が電子から散乱されると, 光子の Δλ=λC(1−cos⁡θ)\Delta \lambda=\lambda_C(1-\cos \theta)

11.9: 光電効果 ​

  1. 光電効果: W+mvmax⁡2/2=hνW+m v_{\max }^2 / 2=h \nu ( WW は仕事関数 )). I−VI-V グラフ：光電流は阻止電圧 V=−(hν−W)/eV=-(h \nu-W) / e で始まり, 正方向に電圧が大きくなると緩和する.

11.10: Stefan-Boltzmann の法則 ​

  1. Stefan-Boltzmann の法則 : P=σAT4P=\sigma A T^4
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/12.html b/academic/physics/ipho-formulas-jpn/12.html index 559be5c7..2d33186d 100644 --- a/academic/physics/ipho-formulas-jpn/12.html +++ b/academic/physics/ipho-formulas-jpn/12.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 12 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 12 ​

12: Kepler の法則 ​

12.1: F & U ​

  1. F=GMm/r2,U=−GMm/r. F=G M m / r^2, U=-G M m / r \text {. }

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

  1. Kepler の第三法則 :r−2: r^{-2} に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の 32\frac{3}{2} 乗 に比例する :

    T12/T22=a13/a23T_1^2 / T_2^2=a_1^3 / a_2^3

12.5: 楕円軌道 ​

  1. 重力場中で楕円軌道を描く質点の全エネルギー (K+U)(K+U) :

    E=−GMm/2aE=-G M m / 2 a

12.6: 楕円率 ​

  1. 楕円率が ε=d/a≪1\varepsilon=d / a \ll 1 の場合, 軌道は焦点をずらし た円形をしていると考えられる.

12.7: 楕円の性質 ​

  1. 楕円の性質 :l1+l2=2a（l1: l_1+l_2=2 a （ l_1 と l2l_2 は焦点までの距 離). α1=α2\alpha_1=\alpha_2 (一方の焦点から出た光は他方の焦点に 反射する). S=πabS=\pi a b.

12.8: 円の中心 ​

  1. 円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.

12.9: nge-Lenz ベクトル ​

  1. Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 ee に一致する.]：

    e=v×MGMm−er= const. \boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. }

- +
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 12 ​

12: Kepler の法則 ​

12.1: F & U ​

  1. F=GMm/r2,U=−GMm/r. F=G M m / r^2, U=-G M m / r \text {. }

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

  1. Kepler の第三法則 :r−2: r^{-2} に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の 32\frac{3}{2} 乗 に比例する :

    T12/T22=a13/a23T_1^2 / T_2^2=a_1^3 / a_2^3

12.5: 楕円軌道 ​

  1. 重力場中で楕円軌道を描く質点の全エネルギー (K+U)(K+U) :

    E=−GMm/2aE=-G M m / 2 a

12.6: 楕円率 ​

  1. 楕円率が ε=d/a≪1\varepsilon=d / a \ll 1 の場合, 軌道は焦点をずらし た円形をしていると考えられる.

12.7: 楕円の性質 ​

  1. 楕円の性質 :l1+l2=2a（l1: l_1+l_2=2 a （ l_1 と l2l_2 は焦点までの距 離). α1=α2\alpha_1=\alpha_2 (一方の焦点から出た光は他方の焦点に 反射する). S=πabS=\pi a b.

12.8: 円の中心 ​

  1. 円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.

12.9: nge-Lenz ベクトル ​

  1. Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 ee に一致する.]：

    e=v×MGMm−er= const. \boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. }

+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/13.html b/academic/physics/ipho-formulas-jpn/13.html index 3606d0a1..dd6495e5 100644 --- a/academic/physics/ipho-formulas-jpn/13.html +++ b/academic/physics/ipho-formulas-jpn/13.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 13 | Toshiki's Note - + - + @@ -92,9 +92,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​

13.11: 電場と磁場の Lorentz 変換 ​

  1. 電場と磁場の Lorentz 変換 : E∥′=E∥,B∥′=B∥\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|},

    E⊥′/c=γ(E⊥/c+v/c×B⊥),B⊥′=γ(B⊥−v/c×E⊥/c)\begin{gathered} \boldsymbol{E}_{\perp}^{\prime} / c=\gamma\left(\boldsymbol{E}_{\perp} / c+\boldsymbol{v} / c \times \boldsymbol{B}_{\perp}\right), \\ \boldsymbol{B}_{\perp}^{\prime}=\gamma\left(\boldsymbol{B}_{\perp}-\boldsymbol{v} / c \times \boldsymbol{E}_{\perp} / c\right) \end{gathered}

- +M1001 80h400000v40h-400000z">​

13.11: 電場と磁場の Lorentz 変換 ​

  1. 電場と磁場の Lorentz 変換 : E∥′=E∥,B∥′=B∥\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|},

    E⊥′/c=γ(E⊥/c+v/c×B⊥),B⊥′=γ(B⊥−v/c×E⊥/c)\begin{gathered} \boldsymbol{E}_{\perp}^{\prime} / c=\gamma\left(\boldsymbol{E}_{\perp} / c+\boldsymbol{v} / c \times \boldsymbol{B}_{\perp}\right), \\ \boldsymbol{B}_{\perp}^{\prime}=\gamma\left(\boldsymbol{B}_{\perp}-\boldsymbol{v} / c \times \boldsymbol{E}_{\perp} / c\right) \end{gathered}

+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/2.html b/academic/physics/ipho-formulas-jpn/2.html index 82c569a1..d8d6f30a 100644 --- a/academic/physics/ipho-formulas-jpn/2.html +++ b/academic/physics/ipho-formulas-jpn/2.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 2 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 2 ​

2: 一般的な推奨事 ​

  1. 全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または ∞\infty ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.
  2. 問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む.
  3. 長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする.
  4. 絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける.
  5. 実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.
- +
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 2 ​

2: 一般的な推奨事 ​

  1. 全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または ∞\infty ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.
  2. 問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む.
  3. 長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする.
  4. 絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける.
  5. 実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/3.html b/academic/physics/ipho-formulas-jpn/3.html index 5646b9b7..3679d05d 100644 --- a/academic/physics/ipho-formulas-jpn/3.html +++ b/academic/physics/ipho-formulas-jpn/3.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 3 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 3 ​

3: 運動学 ​

3.1: 質点 ​

  1. 質点または剛体の並進運動の場合（積分 → グラフの下 の面積）：

    v=dxdt,x=∫vdt(x=∫vx dt など )a=dvdt=d2xdt2,v=∫adtt=∫vx−1 dx=∫ax−1 dvx,x=∫vxax dvx\begin{gathered} \boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d} t}, \boldsymbol{x}=\int \boldsymbol{v} \mathrm{d} t\left(x=\int v_x \mathrm{~d} t \text { など }\right) \\ \boldsymbol{a}=\frac{\mathrm{d} \boldsymbol{v}}{\mathrm{d} t}=\frac{\mathrm{d}^2 \boldsymbol{x}}{\mathrm{d} t^2}, \boldsymbol{v}=\int \boldsymbol{a} \mathrm{d} t \\ t=\int v_x^{-1} \mathrm{~d} x=\int a_x^{-1} \mathrm{~d} v_x, x=\int \frac{v_x}{a_x} \mathrm{~d} v_x \end{gathered}

    もし aa が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば

    x=v0t+at2/2=(v2−v02)/2a. x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. }

3.2: 回転運動 ​

  1. 回転運動は, 並進運動と似ていて：

    ω=dφ/dt,ε=dω/dta=τdv/dt+nv2/R\begin{aligned} \omega & =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} & =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned}

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

  1. 剛体の運動:
    • vAcos⁡α=vBcos⁡βv_A \cos \alpha=v_B \cos \beta ここで, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B は剛体上の点 AA と BB の速度, α\alpha と β\beta は vA\boldsymbol{v}_A と vB\boldsymbol{v}_B が直線 ABA B となす角.
    • 瞬間回転中心 (#質点の軌道 の曲率中心）は, a\boldsymbol{a} と b\boldsymbol{b} に下ろした垂線の交点. 又は もし vA,vB⊥AB\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B ならば, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B の先端を結ぶ 直線と ABA B の交点.

3.5: 非慣性系 ​

  1. 非慣性系：

    v2=v0+v1,a2=a0+a1+ω2R+aCor ここで, aCor⊥v1. もし v1=0 なら aCor=0.\begin{array}{r} \quad \boldsymbol{v}_2=\boldsymbol{v}_0+\boldsymbol{v}_1, \boldsymbol{a}_2=\boldsymbol{a}_0+\boldsymbol{a}_1+\omega^2 \boldsymbol{R}+\boldsymbol{a}_{C o r} \\ \text { ここで, } \boldsymbol{a}_{C o r} \perp \boldsymbol{v}_1 . \text { もし } \boldsymbol{v}_1=0 \text { なら } \boldsymbol{a}_{C o r}=0 . \end{array}

3.6: 弾道問題 ​

  1. 弾道問題：到達可能な範囲は

    y≤v02/(2g)−gx2/(2v02)y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right)

    最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
- +
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 3 ​

3: 運動学 ​

3.1: 質点 ​

  1. 質点または剛体の並進運動の場合（積分 → グラフの下 の面積）：

    v=dxdt,x=∫vdt(x=∫vx dt など )a=dvdt=d2xdt2,v=∫adtt=∫vx−1 dx=∫ax−1 dvx,x=∫vxax dvx\begin{gathered} \boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{x}}{\mathrm{d} t}, \boldsymbol{x}=\int \boldsymbol{v} \mathrm{d} t\left(x=\int v_x \mathrm{~d} t \text { など }\right) \\ \boldsymbol{a}=\frac{\mathrm{d} \boldsymbol{v}}{\mathrm{d} t}=\frac{\mathrm{d}^2 \boldsymbol{x}}{\mathrm{d} t^2}, \boldsymbol{v}=\int \boldsymbol{a} \mathrm{d} t \\ t=\int v_x^{-1} \mathrm{~d} x=\int a_x^{-1} \mathrm{~d} v_x, x=\int \frac{v_x}{a_x} \mathrm{~d} v_x \end{gathered}

    もし aa が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば

    x=v0t+at2/2=(v2−v02)/2a. x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. }

3.2: 回転運動 ​

  1. 回転運動は, 並進運動と似ていて：

    ω=dφ/dt,ε=dω/dta=τdv/dt+nv2/R\begin{aligned} \omega & =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} & =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned}

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

  1. 剛体の運動:
    • vAcos⁡α=vBcos⁡βv_A \cos \alpha=v_B \cos \beta ここで, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B は剛体上の点 AA と BB の速度, α\alpha と β\beta は vA\boldsymbol{v}_A と vB\boldsymbol{v}_B が直線 ABA B となす角.
    • 瞬間回転中心 (#質点の軌道 の曲率中心）は, a\boldsymbol{a} と b\boldsymbol{b} に下ろした垂線の交点. 又は もし vA,vB⊥AB\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B ならば, vA\boldsymbol{v}_A と vB\boldsymbol{v}_B の先端を結ぶ 直線と ABA B の交点.

3.5: 非慣性系 ​

  1. 非慣性系：

    v2=v0+v1,a2=a0+a1+ω2R+aCor ここで, aCor⊥v1. もし v1=0 なら aCor=0.\begin{array}{r} \quad \boldsymbol{v}_2=\boldsymbol{v}_0+\boldsymbol{v}_1, \boldsymbol{a}_2=\boldsymbol{a}_0+\boldsymbol{a}_1+\omega^2 \boldsymbol{R}+\boldsymbol{a}_{C o r} \\ \text { ここで, } \boldsymbol{a}_{C o r} \perp \boldsymbol{v}_1 . \text { もし } \boldsymbol{v}_1=0 \text { なら } \boldsymbol{a}_{C o r}=0 . \end{array}

3.6: 弾道問題 ​

  1. 弾道問題：到達可能な範囲は

    y≤v02/(2g)−gx2/(2v02)y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right)

    最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/4.html b/academic/physics/ipho-formulas-jpn/4.html index 4fd94370..bbd821b8 100644 --- a/academic/physics/ipho-formulas-jpn/4.html +++ b/academic/physics/ipho-formulas-jpn/4.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 4 | Toshiki's Note - + - + @@ -41,9 +41,9 @@ c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z -M1001 80h400000v40h-400000z">​,l~=l+Ic​/ml​

4.12: 慣性モーメントの係数 ​

  1. 慣性モーメントの係数 : 円柱 12\frac{1}{2}, 球 25\frac{2}{5}, 球殼 23\frac{2}{3}, 棒 112\frac{1}{12} （端に対しては 13\frac{1}{3} ), 正方形 16\frac{1}{6}.

4.13: よく使われる保存則 ​

  1. よく使われる保存則：エネルギー（弾性衝突，摩擦な し), 運動量（正味の外力なし, 各方向について成立), 角運動量（正味の外トルクなし, 例えば, 外力のうでの 長さが 0 (これが 2 又は 3 点のまわりに成り立てば運 動量保存で代用できる))

4.14: 非慣性系における見かけの力 ​

  1. 非慣性系における見かけの力 : 慣性力 −ma-m a, 遠心力 mω2Rm \omega^2 \boldsymbol{R}, Coriolis 力 2mv×Ω2 m \boldsymbol{v} \times \Omega (避けた方がよい. 速 度に垂直なので仕事はしない).

4.15: 傾いた座標 ​

  1. 傾いた座標：斜面上での運動については, 斜面に平行 と垂直な方向に軸をとるのがよい。このとき重力加速 度は xx 成分と yy 成分をもつ. 軸は斜交することもある が, v=vxex+vyey\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y のとき vxv_x は v\boldsymbol{v} の xx 軸への射 影ではない.

4.16: 2 つの物体の衝突 ​

  1. 2 つの物体の衝突 : 保存されるのは, a) 全運動量, b) 全角運動量，c) 一方の物体の衝突点に関する角運動量, d) 全エネルギー（弾性衝突の場合， 摩擦がある場合 は, 摩擦力に垂直な方向の運動エネルギーが保存される. e) 衝突中に滑りが止まったならば，接触点の最終 速度は接触面上にある. f) 滑りが止まらなかったなら ば, 一方の物体から他方に伝わる運動量は, 接触面の 法線と arctan⁡μ\arctan \mu の角度をなす.

4.17: 剛体のすべての運動 ​

  1. 剛体のすべての運動は (物体の各点の速度を見ると) 瞬 間回転中心 CC まわりの回転として表せる. 物体上の点 PP の CC からの距離は PP の軌跡の曲率半径とは異なる ことに注意せよ.

4.18: 紐の張力 ​

  1. 紐の張力：重さのある吊り紐では, 張力の水平成分は 一定で垂直成分は下にある紐の重さにより変わる. 滑 らかな面の上の紐による（単位長さあたりの）力は，そ の曲率半径と張力で決まり, N=T/RN=T / R. 似た場合と して, 表面張力による圧力は p=2σ/Rp=2 \sigma / R. 導出には直 径に沿った圧力を調ベる.

4.19: 液体の表面 ​

  1. 液体の表面は（表面張力を無視すれば）等ポテンシャ ル面になる. 非圧縮性流体では, ww をポテンシャルエ ネルギーの体積密度として, p=P0−wp=P_0-w.

4.20: 非圧縮性流体に対する Bernoulli の法則 ​

  1. 非圧縮性流体に対する Bernoulli の法則：

    p+12ρv2+ρϕ= const. p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. }

    一様な重力場では ϕ=gh\phi=g h. 比熱が cp[ J/kg]c_p[\mathrm{~J} / \mathrm{kg}] である気 体では,

    12v2+cpT= const. \frac{1}{2} v^2+c_p T=\text { const. }

4.21: 直線的な流線 ​

  1. 直線的な流線に沾う運動量の連続性 : p+ρv2=p+\rho v^2= const.

4.22: 断熱不変量 ​

  1. 断熱不変量 : 振動する系の 1 周期の間のパラメータの 相対的な変化が小さければ，位相空間（ x−px-p 座標で表さ れる）上に書かれるループの面積は非常に高い精度で 保存される.

4.23: 安定性 ​

  1. 安定性を調ベるには, a) ポテンシャルエネルギー最小 の原理，又は b) 仮想仕事の原理を用いる.

4.24: 空間的に有限な運動に対する Virial 定理 ​

  1. 空間的に有限な運動に対する Virial 定理：a) もし F∝∣r∣F \propto|\boldsymbol{r}| ならば ⟨K⟩=⟨U⟩\langle K\rangle=\langle U\rangle (時間平均). b) もし F∝∣r∣−2F \propto|\boldsymbol{r}|^{-2} ならば 2⟨K⟩=−⟨U⟩2\langle K\rangle=-\langle U\rangle.

4.25: Tsiolkovsky 公式 ​

  1. Tsiolkovsky の公式（ロケット）: Δv=uln⁡Mm\Delta v=u \ln \frac{M}{m}
- +M1001 80h400000v40h-400000z">​,l~=l+Ic​/ml​

4.12: 慣性モーメントの係数 ​

  1. 慣性モーメントの係数 : 円柱 12\frac{1}{2}, 球 25\frac{2}{5}, 球殼 23\frac{2}{3}, 棒 112\frac{1}{12} （端に対しては 13\frac{1}{3} ), 正方形 16\frac{1}{6}.

4.13: よく使われる保存則 ​

  1. よく使われる保存則：エネルギー（弾性衝突，摩擦な し), 運動量（正味の外力なし, 各方向について成立), 角運動量（正味の外トルクなし, 例えば, 外力のうでの 長さが 0 (これが 2 又は 3 点のまわりに成り立てば運 動量保存で代用できる))

4.14: 非慣性系における見かけの力 ​

  1. 非慣性系における見かけの力 : 慣性力 −ma-m a, 遠心力 mω2Rm \omega^2 \boldsymbol{R}, Coriolis 力 2mv×Ω2 m \boldsymbol{v} \times \Omega (避けた方がよい. 速 度に垂直なので仕事はしない).

4.15: 傾いた座標 ​

  1. 傾いた座標：斜面上での運動については, 斜面に平行 と垂直な方向に軸をとるのがよい。このとき重力加速 度は xx 成分と yy 成分をもつ. 軸は斜交することもある が, v=vxex+vyey\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y のとき vxv_x は v\boldsymbol{v} の xx 軸への射 影ではない.

4.16: 2 つの物体の衝突 ​

  1. 2 つの物体の衝突 : 保存されるのは, a) 全運動量, b) 全角運動量，c) 一方の物体の衝突点に関する角運動量, d) 全エネルギー（弾性衝突の場合， 摩擦がある場合 は, 摩擦力に垂直な方向の運動エネルギーが保存される. e) 衝突中に滑りが止まったならば，接触点の最終 速度は接触面上にある. f) 滑りが止まらなかったなら ば, 一方の物体から他方に伝わる運動量は, 接触面の 法線と arctan⁡μ\arctan \mu の角度をなす.

4.17: 剛体のすべての運動 ​

  1. 剛体のすべての運動は (物体の各点の速度を見ると) 瞬 間回転中心 CC まわりの回転として表せる. 物体上の点 PP の CC からの距離は PP の軌跡の曲率半径とは異なる ことに注意せよ.

4.18: 紐の張力 ​

  1. 紐の張力：重さのある吊り紐では, 張力の水平成分は 一定で垂直成分は下にある紐の重さにより変わる. 滑 らかな面の上の紐による（単位長さあたりの）力は，そ の曲率半径と張力で決まり, N=T/RN=T / R. 似た場合と して, 表面張力による圧力は p=2σ/Rp=2 \sigma / R. 導出には直 径に沿った圧力を調ベる.

4.19: 液体の表面 ​

  1. 液体の表面は（表面張力を無視すれば）等ポテンシャ ル面になる. 非圧縮性流体では, ww をポテンシャルエ ネルギーの体積密度として, p=P0−wp=P_0-w.

4.20: 非圧縮性流体に対する Bernoulli の法則 ​

  1. 非圧縮性流体に対する Bernoulli の法則：

    p+12ρv2+ρϕ= const. p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. }

    一様な重力場では ϕ=gh\phi=g h. 比熱が cp[ J/kg]c_p[\mathrm{~J} / \mathrm{kg}] である気 体では,

    12v2+cpT= const. \frac{1}{2} v^2+c_p T=\text { const. }

4.21: 直線的な流線 ​

  1. 直線的な流線に沾う運動量の連続性 : p+ρv2=p+\rho v^2= const.

4.22: 断熱不変量 ​

  1. 断熱不変量 : 振動する系の 1 周期の間のパラメータの 相対的な変化が小さければ，位相空間（ x−px-p 座標で表さ れる）上に書かれるループの面積は非常に高い精度で 保存される.

4.23: 安定性 ​

  1. 安定性を調ベるには, a) ポテンシャルエネルギー最小 の原理，又は b) 仮想仕事の原理を用いる.

4.24: 空間的に有限な運動に対する Virial 定理 ​

  1. 空間的に有限な運動に対する Virial 定理：a) もし F∝∣r∣F \propto|\boldsymbol{r}| ならば ⟨K⟩=⟨U⟩\langle K\rangle=\langle U\rangle (時間平均). b) もし F∝∣r∣−2F \propto|\boldsymbol{r}|^{-2} ならば 2⟨K⟩=−⟨U⟩2\langle K\rangle=-\langle U\rangle.

4.25: Tsiolkovsky 公式 ​

  1. Tsiolkovsky の公式（ロケット）: Δv=uln⁡Mm\Delta v=u \ln \frac{M}{m}
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/5.html b/academic/physics/ipho-formulas-jpn/5.html index d82cca01..efdf82db 100644 --- a/academic/physics/ipho-formulas-jpn/5.html +++ b/academic/physics/ipho-formulas-jpn/5.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 5 | Toshiki's Note - + - + @@ -101,9 +101,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​.

5.10: Doppler 効果 ​

  1. Doppler 効果 : ν=ν01+v∥/cs1−u∥/cs\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}.

5.11: Huygens の原理 ​

  1. Huygens の原理 : 波面は段階的に構成される. 過去 の波面のすべての点に仮想的な波源を置く. 結果は距 離 Δx=csΔt\Delta x=c_s \Delta t で区切られた曲線（ここで Δt\Delta t は時間 間隔， csc_s は与えられた点の速度). 波は波面に垂直に 進む.
- +M1001 80h400000v40h-400000z">​.

5.10: Doppler 効果 ​

  1. Doppler 効果 : ν=ν01+v∥/cs1−u∥/cs\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}.

5.11: Huygens の原理 ​

  1. Huygens の原理 : 波面は段階的に構成される. 過去 の波面のすべての点に仮想的な波源を置く. 結果は距 離 Δx=csΔt\Delta x=c_s \Delta t で区切られた曲線（ここで Δt\Delta t は時間 間隔， csc_s は与えられた点の速度). 波は波面に垂直に 進む.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/6.html b/academic/physics/ipho-formulas-jpn/6.html index 4b32f5c2..28098221 100644 --- a/academic/physics/ipho-formulas-jpn/6.html +++ b/academic/physics/ipho-formulas-jpn/6.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 6 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 6 ​

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

  1. Fermat の原理 : 点 AA から BB への波の経路は波の移動 時間が最も短いもの.

6.2: Snell 法則 ​

  1. Snell の法則 :

sin⁡α1/sin⁡α2=n2/n1=v1/v2.\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 .

6.3: 屈折率 ​

  1. 屈折率が連続的に変化するならば，媒質を屈折率が nn で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, n′=n/r(rn^{\prime}=n / r \quad(r は曲率半径 ))

6.4: 屈折率な座標 ​

  1. 屈折率が z\mathrm{z} 座標にのみ依存するならば, 光子の運動量 px,pyp_x, p_y とエネルギーは保存される:

kx,ky= const., ∣k∣/n= const. k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. }

6.5:薄いレンズの式 ​

  1. 薄いレンズの式（符号に注意する）：

1/a+1/b=1/f≡D1 / a+1 / b=1 / f \equiv D

6.6: Newton の式 ​

  1. Newton の式 : 物体側焦点から物体までの距離を x1x_1, 像側焦点から像までの距離を x2x_2 とすると, x1x2=f2x_1 x_2=f^2

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

  1. 光束 Φ\Phi [単位: lumen (lm⁡)(\operatorname{lm})] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, I=Φ/ΩI=\Phi / \Omega. 照度 [lux⁡(lx)][\operatorname{lux}(\mathrm{lx})] は（面に入射する） 面積あたりの光束で, E=Φ/SE=\Phi / S.

6.10: Gauss 定理 ​

  1. 光束についての Gauss の定理 : 光度 IiI_i の点光源を囲 む閉曲面を通って外に出る光束は, Φ=4π∑Ii\Phi=4 \pi \sum I_i. 光 源が 1 つで距離が rr のとき E=I/r2E=I / r^2

6.11: 実験のヒント ​

  1. 実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.
- +
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 6 ​

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

  1. Fermat の原理 : 点 AA から BB への波の経路は波の移動 時間が最も短いもの.

6.2: Snell 法則 ​

  1. Snell の法則 :

sin⁡α1/sin⁡α2=n2/n1=v1/v2.\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 .

6.3: 屈折率 ​

  1. 屈折率が連続的に変化するならば，媒質を屈折率が nn で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, n′=n/r(rn^{\prime}=n / r \quad(r は曲率半径 ))

6.4: 屈折率な座標 ​

  1. 屈折率が z\mathrm{z} 座標にのみ依存するならば, 光子の運動量 px,pyp_x, p_y とエネルギーは保存される:

kx,ky= const., ∣k∣/n= const. k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. }

6.5:薄いレンズの式 ​

  1. 薄いレンズの式（符号に注意する）：

1/a+1/b=1/f≡D1 / a+1 / b=1 / f \equiv D

6.6: Newton の式 ​

  1. Newton の式 : 物体側焦点から物体までの距離を x1x_1, 像側焦点から像までの距離を x2x_2 とすると, x1x2=f2x_1 x_2=f^2

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

  1. 光束 Φ\Phi [単位: lumen (lm⁡)(\operatorname{lm})] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, I=Φ/ΩI=\Phi / \Omega. 照度 [lux⁡(lx)][\operatorname{lux}(\mathrm{lx})] は（面に入射する） 面積あたりの光束で, E=Φ/SE=\Phi / S.

6.10: Gauss 定理 ​

  1. 光束についての Gauss の定理 : 光度 IiI_i の点光源を囲 む閉曲面を通って外に出る光束は, Φ=4π∑Ii\Phi=4 \pi \sum I_i. 光 源が 1 つで距離が rr のとき E=I/r2E=I / r^2

6.11: 実験のヒント ​

  1. 実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/7.html b/academic/physics/ipho-formulas-jpn/7.html index 3f392168..91b08990 100644 --- a/academic/physics/ipho-formulas-jpn/7.html +++ b/academic/physics/ipho-formulas-jpn/7.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 7 | Toshiki's Note - + - + @@ -42,9 +42,9 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5, c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z -M1001 80h400000v40h-400000z">​ (普通 μ≈1\mu \approx 1 ) であることに注意せよ. エネ ルギー流密度（単位面積を通過する単位時間あたりの エネルギー）: I=cnε0E2=cnμ0B2(EI=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E と BB は実 効值)

7.12: Malus の法則 ​

  1. Malus の法則 : 直線偏光が角度 φ\varphi で偏光板を通過する と I=I0cos⁡2φI=I_0 \cos ^2 \varphi

7.13: 1/4 波長版 ​

  1. 1/41 / 4 波長版 : 直線偏光成分間の位相が π/2\pi / 2 ずれる.

7.14: Brewster 角 ​

  1. Brewster 角: 入射角が tan⁡φ=n\tan \varphi=n を満たすとき, 反 射波と屈折波が垂直になり反射波は直線偏光となる.

7.15: 光学素子による回折 ​

  1. 光学素子による回折 : レンズやプリズムなどを通る光 の光学距離を計算する必要はなく, 図形的に考える. 例えば，双プリズムは二重スリットによる回折と同じ 回折をする.

7.16: 光ファイバー ​

  1. 光ファイバー：Mach-Zehnder 干渉計は二重スリッ トによる干渉と, 円形共振器は Fabry-Pérot 干渉計 と似ている. Bragg フィルターは X\mathrm{X} 線の場合と同 じように働く. シングルモードの光ファイバーでは, Δn/n≈12(λ/d)2\Delta n / n \approx \frac{1}{2}(\lambda / d)^2.
- +M1001 80h400000v40h-400000z">​ (普通 μ≈1\mu \approx 1 ) であることに注意せよ. エネ ルギー流密度（単位面積を通過する単位時間あたりの エネルギー）: I=cnε0E2=cnμ0B2(EI=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E と BB は実 効值)

7.12: Malus の法則 ​

  1. Malus の法則 : 直線偏光が角度 φ\varphi で偏光板を通過する と I=I0cos⁡2φI=I_0 \cos ^2 \varphi

7.13: 1/4 波長版 ​

  1. 1/41 / 4 波長版 : 直線偏光成分間の位相が π/2\pi / 2 ずれる.

7.14: Brewster 角 ​

  1. Brewster 角: 入射角が tan⁡φ=n\tan \varphi=n を満たすとき, 反 射波と屈折波が垂直になり反射波は直線偏光となる.

7.15: 光学素子による回折 ​

  1. 光学素子による回折 : レンズやプリズムなどを通る光 の光学距離を計算する必要はなく, 図形的に考える. 例えば，双プリズムは二重スリットによる回折と同じ 回折をする.

7.16: 光ファイバー ​

  1. 光ファイバー：Mach-Zehnder 干渉計は二重スリッ トによる干渉と, 円形共振器は Fabry-Pérot 干渉計 と似ている. Bragg フィルターは X\mathrm{X} 線の場合と同 じように働く. シングルモードの光ファイバーでは, Δn/n≈12(λ/d)2\Delta n / n \approx \frac{1}{2}(\lambda / d)^2.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/8.html b/academic/physics/ipho-formulas-jpn/8.html index 58c4dfc0..6ddaeb34 100644 --- a/academic/physics/ipho-formulas-jpn/8.html +++ b/academic/physics/ipho-formulas-jpn/8.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 8 | Toshiki's Note - + - + @@ -54,9 +54,9 @@ c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z -M834 80h400000v40h-400000z">​ が成立.
- +M834 80h400000v40h-400000z">​ が成立.
+ \ No newline at end of file diff --git a/academic/physics/ipho-formulas-jpn/9.html b/academic/physics/ipho-formulas-jpn/9.html index 2deec739..a5783df2 100644 --- a/academic/physics/ipho-formulas-jpn/9.html +++ b/academic/physics/ipho-formulas-jpn/9.html @@ -5,12 +5,12 @@ Formulas for IPhO 日本語版: Section 9 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 9 ​

9: 電磁気学 ​

9.1: Coulomb の法則 ​

  1. F=kq1q2/r2,U=kq1q2/rF=k q_1 q_2 / r^2, U=k q_1 q_2 / r で, Kepler の法則が 使える (Section 12 参照).

9.2: Gauss の法則 ​

  1. Gauss の法則 : ∮B⋅dS=0\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0,

    ∮εE⋅dS=Q,∮g⋅dS=−4πGM\oint \varepsilon \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{S}=Q, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{S}=-4 \pi G M

9.3: 循環定理 ​

  1. 循環定理 :

    ∮E⋅dl=0(=Φ˙),∮B⋅dlμ=I,∮g⋅dl=0\oint \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{l}=0(=\dot{\Phi}), \oint \frac{\boldsymbol{B} \cdot \mathrm{d} \boldsymbol{l}}{\mu}=I, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{l}=0

9.4: 電流素片により生じる磁束密度 ​

  1. 電流素片により生じる磁束密度 :

    dB=μI4πdl×err2.\mathrm{d} \boldsymbol{B}=\frac{\mu I}{4 \pi} \frac{\mathrm{d} \boldsymbol{l} \times \boldsymbol{e}_r}{r^2} .

    したがって電流 II が流れる円形回路の中心では B=μ0I2rB=\frac{\mu_0 I}{2 r}.

9.5: ローレンツ力 ​

  1. F=e(E+v×B),F=I×Bl\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l.

9.6: Gauss の定理と循環定理より ​

  1. Gauss の定理と循環定理より：帯電した導線について E=σ2πε0rE=\frac{\sigma}{2 \pi \varepsilon_0 r}, 電流が流れる導線について B=μ0I2πrB=\frac{\mu_0 I}{2 \pi r}. 帯電した面について E=σ2ε0E=\frac{\sigma}{2 \varepsilon_0}, 電流が流れる面につい て B=μ0i2B=\frac{\mu_0 i}{2}. 一様に帯電した球殼（又は無限に長い円 筒）の内部で E=0E=0, 軸に沿って表面に電流が流れる 円筒の内部で B=0B=0. 密度 ρ\rho で一様に帯電, 又は一様 な電流 i\boldsymbol{i} が流れる, 球 (d=3)/(d=3) / 円柱 (d−2)/(d-2) / 平面 (d=1)(d=1) の内部で,

    E=ρεdr,B=1μdi×r\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r}

9.7: 長いソレノイド ​

  1. 長いソレノイド: 内部で B=μnIB=\mu n I, 外部で B=0B=0. 磁束 Φ=NBS(n=Nl)\Phi=N B S\left(n=\frac{N}{l}\right). インダクタンス L=L= Φ/I=μn2V\Phi / I=\mu n^2 V. 短いソレノイド :B∥=μnIΩ4π(Ω: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega は 立体角).

9.8: 磁場を小型コイルや衝撃検流計で測定する ​

  1. 磁場を小型コイルや衝撃検流計で測定する： q=q= ∫VR dt=NSΔB/R\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R.

9.9: 静電場のエネルギー ​

  1. 静電場のエネルギー：

    U=k∑i<jqiqjrij=12∫ϕ(r)dq, dq=ρ(r)dVU=k \sum_{i<j} \frac{q_i q_j}{r_{i j}}=\frac{1}{2} \int \phi(\boldsymbol{r}) \mathrm{d} q, \mathrm{~d} q=\rho(\boldsymbol{r}) \mathrm{d} V

9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​

  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

  1. 全ての電荷が距離 rr にある場合（例えば，不均一に帯 電した球やリングの中心） ϕϕ=kQ/r\phi \phi=k Q / r

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

  1. 導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも QQ という 電荷を持った導電性の球があるように見える).

9.15: 静電容量 ​

  1. 静電容量: C=εS/dC=\varepsilon S / d (平板), 4πεr4 \pi \varepsilon r (球), 2πεl(ln⁡R/r)−12 \pi \varepsilon l(\ln R / r)^{-1} (同軸円筒).

9.16: 双極子モーメント ​

  1. 双極子モーメント：

    pe=∑qiri=qd,pμ=IS\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S}

9.17: 双極子場 ​

  1. 双極子場 : ϕ=kp⋅er/r2,E,B∝r−3\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}

9.18: 双極子に働く力 ​

  1. 双極子に働く力 : F=(pe⋅E)′,F=(pμ⋅B)′F=\left(\boldsymbol{p}_e \cdot \boldsymbol{E}\right)^{\prime}, F=\left(\boldsymbol{p}_\mu \cdot \boldsymbol{B}\right)^{\prime} [訳 者注 : ここの微分はむしろ grad⁡\operatorname{grad} である]. 2 つの双極 子間の相互作用 :F∝r−4: F \propto r^{-4}.

9.19: 磁気双極子としての点電荷 ​

  1. 磁気双極子としての点電荷 : pμ∝Φ∝v⊥2/Bp_\mu \propto \Phi \propto v_{\perp}^2 / B は断熱 不変量 (Section 4: #22 参照).

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

  1. 一様（電）場中の球 (円柱) の分極 : (+ρ(+\rho と −ρ-\rho に一 様に帯電した球 (円柱) の重ね合わせで， d∝Ed \propto E.

9.22: 渦電流 ​

  1. 渦電流: 電流損失密度 ≈B2v2/ρ.1\approx B^2 v^2 / \rho .1 回の通過で与え られる運動量 : Fτ≈B2a3d/ρF \tau \approx B^2 a^3 d / \rho (ここで dd は厚さ, aa は大きさ).
- +
Toshiki's Note

Appearance

TOC

Formulas for IPhO 日本語版: Section 9 ​

9: 電磁気学 ​

9.1: Coulomb の法則 ​

  1. F=kq1q2/r2,U=kq1q2/rF=k q_1 q_2 / r^2, U=k q_1 q_2 / r で, Kepler の法則が 使える (Section 12 参照).

9.2: Gauss の法則 ​

  1. Gauss の法則 : ∮B⋅dS=0\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0,

    ∮εE⋅dS=Q,∮g⋅dS=−4πGM\oint \varepsilon \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{S}=Q, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{S}=-4 \pi G M

9.3: 循環定理 ​

  1. 循環定理 :

    ∮E⋅dl=0(=Φ˙),∮B⋅dlμ=I,∮g⋅dl=0\oint \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{l}=0(=\dot{\Phi}), \oint \frac{\boldsymbol{B} \cdot \mathrm{d} \boldsymbol{l}}{\mu}=I, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{l}=0

9.4: 電流素片により生じる磁束密度 ​

  1. 電流素片により生じる磁束密度 :

    dB=μI4πdl×err2.\mathrm{d} \boldsymbol{B}=\frac{\mu I}{4 \pi} \frac{\mathrm{d} \boldsymbol{l} \times \boldsymbol{e}_r}{r^2} .

    したがって電流 II が流れる円形回路の中心では B=μ0I2rB=\frac{\mu_0 I}{2 r}.

9.5: ローレンツ力 ​

  1. F=e(E+v×B),F=I×Bl\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l.

9.6: Gauss の定理と循環定理より ​

  1. Gauss の定理と循環定理より：帯電した導線について E=σ2πε0rE=\frac{\sigma}{2 \pi \varepsilon_0 r}, 電流が流れる導線について B=μ0I2πrB=\frac{\mu_0 I}{2 \pi r}. 帯電した面について E=σ2ε0E=\frac{\sigma}{2 \varepsilon_0}, 電流が流れる面につい て B=μ0i2B=\frac{\mu_0 i}{2}. 一様に帯電した球殼（又は無限に長い円 筒）の内部で E=0E=0, 軸に沿って表面に電流が流れる 円筒の内部で B=0B=0. 密度 ρ\rho で一様に帯電, 又は一様 な電流 i\boldsymbol{i} が流れる, 球 (d=3)/(d=3) / 円柱 (d−2)/(d-2) / 平面 (d=1)(d=1) の内部で,

    E=ρεdr,B=1μdi×r\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r}

9.7: 長いソレノイド ​

  1. 長いソレノイド: 内部で B=μnIB=\mu n I, 外部で B=0B=0. 磁束 Φ=NBS(n=Nl)\Phi=N B S\left(n=\frac{N}{l}\right). インダクタンス L=L= Φ/I=μn2V\Phi / I=\mu n^2 V. 短いソレノイド :B∥=μnIΩ4π(Ω: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega は 立体角).

9.8: 磁場を小型コイルや衝撃検流計で測定する ​

  1. 磁場を小型コイルや衝撃検流計で測定する： q=q= ∫VR dt=NSΔB/R\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R.

9.9: 静電場のエネルギー ​

  1. 静電場のエネルギー：

    U=k∑i<jqiqjrij=12∫ϕ(r)dq, dq=ρ(r)dVU=k \sum_{i<j} \frac{q_i q_j}{r_{i j}}=\frac{1}{2} \int \phi(\boldsymbol{r}) \mathrm{d} q, \mathrm{~d} q=\rho(\boldsymbol{r}) \mathrm{d} V

9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​

  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

  1. 全ての電荷が距離 rr にある場合（例えば，不均一に帯 電した球やリングの中心） ϕϕ=kQ/r\phi \phi=k Q / r

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

  1. 導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも QQ という 電荷を持った導電性の球があるように見える).

9.15: 静電容量 ​

  1. 静電容量: C=εS/dC=\varepsilon S / d (平板), 4πεr4 \pi \varepsilon r (球), 2πεl(ln⁡R/r)−12 \pi \varepsilon l(\ln R / r)^{-1} (同軸円筒).

9.16: 双極子モーメント ​

  1. 双極子モーメント：

    pe=∑qiri=qd,pμ=IS\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S}

9.17: 双極子場 ​

  1. 双極子場 : ϕ=kp⋅er/r2,E,B∝r−3\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}

9.18: 双極子に働く力 ​

  1. 双極子に働く力 : F=(pe⋅E)′,F=(pμ⋅B)′F=\left(\boldsymbol{p}_e \cdot \boldsymbol{E}\right)^{\prime}, F=\left(\boldsymbol{p}_\mu \cdot \boldsymbol{B}\right)^{\prime} [訳 者注 : ここの微分はむしろ grad⁡\operatorname{grad} である]. 2 つの双極 子間の相互作用 :F∝r−4: F \propto r^{-4}.

9.19: 磁気双極子としての点電荷 ​

  1. 磁気双極子としての点電荷 : pμ∝Φ∝v⊥2/Bp_\mu \propto \Phi \propto v_{\perp}^2 / B は断熱 不変量 (Section 4: #22 参照).

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

  1. 一様（電）場中の球 (円柱) の分極 : (+ρ(+\rho と −ρ-\rho に一 様に帯電した球 (円柱) の重ね合わせで， d∝Ed \propto E.

9.22: 渦電流 ​

  1. 渦電流: 電流損失密度 ≈B2v2/ρ.1\approx B^2 v^2 / \rho .1 回の通過で与え られる運動量 : Fτ≈B2a3d/ρF \tau \approx B^2 a^3 d / \rho (ここで dd は厚さ, aa は大きさ).
+ \ No newline at end of file diff --git a/academic/vocabulary/2023/02/2023-02-27.html b/academic/vocabulary/2023/02/2023-02-27.html index 596158fd..f3718ed7 100644 --- a/academic/vocabulary/2023/02/2023-02-27.html +++ b/academic/vocabulary/2023/02/2023-02-27.html @@ -5,12 +5,12 @@ 2023-2-27: Vocabulary | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

2023-2-27: Vocabulary ​

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

- +
Toshiki's Note

Appearance

TOC

2023-2-27: Vocabulary ​

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

+ \ No newline at end of file diff --git a/academic/vocabulary/index.html b/academic/vocabulary/index.html index de4d2c96..86dc9ae6 100644 --- a/academic/vocabulary/index.html +++ b/academic/vocabulary/index.html @@ -5,12 +5,12 @@ Welcome to My Vocabulary List! | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Welcome to My Vocabulary List! ​

- +
Toshiki's Note

Appearance

TOC

Welcome to My Vocabulary List! ​

+ \ No newline at end of file diff --git a/application/markdown-it-katex/how-to-use.html b/application/markdown-it-katex/how-to-use.html index 9f89fe45..92c6ab23 100644 --- a/application/markdown-it-katex/how-to-use.html +++ b/application/markdown-it-katex/how-to-use.html @@ -5,12 +5,12 @@ @andatoshiki/markdown-it-katex | Toshiki's Note - + - + @@ -32,13 +32,13 @@ -
Toshiki's Note

Appearance

TOC

@andatoshiki/markdown-it-katex ​

Add graceful KaTeX\KaTeX rendering to your Markdown like a charm with markdown-it plugin.

1: Installation ​

Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.

sh
$ npm install markdown-it --save
$ npm install markdown-it --save

First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.

sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
+    
Toshiki's Note
Appearance
TOC
@andatoshiki/markdown-it-katex ​
Add graceful KaTeX\KaTeX rendering to your Markdown like a charm with markdown-it plugin.
1: Installation ​
Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.
sh
$ npm install markdown-it --save
$ npm install markdown-it --save
First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.
sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
 </body>
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
-</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
+</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
 @import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
// ... your other styles
-@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
+@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
     mk = require('andatoshiki/markdown-it-katex')
 
 md.use(mk)
@@ -81,9 +81,9 @@ M834 80h400000v40h-400000z">$$
$$
 \frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right)
 = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\}
-$$
the above block equation renders the LaTeX equation as a block with output as follows,
∂r∂ωr(yωω)=(yωω){(log⁡y)r+∑i=1r(−1)Ir⋯(r−i+1)(log⁡y)riωi}\frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right) = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\} 
5: Syntax ​
Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,
Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.
  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.
6: Supported math syntax ​
KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.
Note
Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.

-    
+$$

the above block equation renders the LaTeX equation as a block with output as follows,

∂r∂ωr(yωω)=(yωω){(log⁡y)r+∑i=1r(−1)Ir⋯(r−i+1)(log⁡y)riωi}\frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right) = \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^ Ir \cdots (r-i+1) (\log y)^{ri}} {\omega^i} \right\}

5: Syntax ​

Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,

Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.

  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.

6: Supported math syntax ​

KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.

Note

Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.

+ \ No newline at end of file diff --git a/application/markdown-it-katex/support-function.html b/application/markdown-it-katex/support-function.html index ade20720..0c55c5ef 100644 --- a/application/markdown-it-katex/support-function.html +++ b/application/markdown-it-katex/support-function.html @@ -5,12 +5,12 @@ KaTeX: Supported functions | Toshiki's Note - + - + @@ -330,9 +330,9 @@ M40 281 V448H0V74H40V241H400000v40z">​ \xmapsto{abc}

|=abc\xlongequal{abc} \xlongequal{abc}

Extensible arrows all can take an optional argument in the same manner, as \xrightarrow[under]{over}.

11: Style, Color, Size, and Font ​

Class Assignment

Color

F=ma\color{blue} F=ma \color{blue} F=ma

Note that KaTeX \color acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:

For color definition, KaTeX color functions will accept the standard HTML predefined color names. They will also accept an RGB argument in CSS hexa­decimal style. The "#" is optional before a six-digit specification.

Font

Ab0\mathrm{Ab0} \mathrm{Ab0}Ab0\mathbf{Ab0} \mathbf{Ab0}Ab\mathit{Ab} \mathit{Ab}
Ab0\mathnormal{Ab0} \mathnormal{Ab0}Ab0\textbf{Ab0} \textbf{Ab0}Ab\textit{Ab} \textit{Ab}
Ab0\textrm{Ab0} \textrm{Ab0}Ab0\bf Ab0 \bf Ab0Ab\it Ab \it Ab
Ab0\rm Ab0 \rm Ab0Ab0\bold{Ab0} \bold{Ab0}AB\Bbb{AB} \Bbb{AB}
Ab0\textnormal{Ab0} \textnormal{Ab0}Ab\boldsymbol{Ab} \boldsymbol{Ab}AB\mathbb{AB} \mathbb{AB}
Ab0\text{Ab0} \text{Ab0}Ab\bm{Ab} \bm{Ab}Ab0\frak{Ab0} \frak{Ab0}
Ab0\mathsf{Ab0} \mathsf{Ab0}Ab0\mathtt{Ab0} \mathtt{Ab0}Ab0\mathfrak{Ab0} \mathfrak{Ab0}
Ab0\textsf{Ab0} \textsf{Ab0}Ab0\texttt{Ab0} \texttt{Ab0}AB0\mathcal{AB0} \mathcal{AB0}
Ab0\sf Ab0 \sf Ab0Ab0\tt Ab0 \tt Ab0AB\mathscr{AB} \mathscr{AB}

One can stack font family, font weight, and font shape by using the \textXX versions of the font functions. So \textsf{\textbf{H}} will produce H\textsf{\textbf{H}}. The other versions do not stack, e.g., \mathsf{\mathbf{H}} will produce H\mathsf{\mathbf{H}}.

In cases where KaTeX fonts do not have a bold glyph, \pmb can simulate one. For example, \pmb{\mu} renders as : μ\pmb{\mu}

Size

AB\Huge AB \Huge ABAB\normalsize AB \normalsize AB
AB\huge AB \huge ABAB\small AB \small AB
AB\LARGE AB \LARGE ABAB\footnotesize AB \footnotesize AB
AB\Large AB \Large ABAB\scriptsize AB \scriptsize AB
AB\large AB \large ABAB\tiny AB \tiny AB

Style

|∑i=1n\displaystyle\sum_{i=1}^n \displaystyle\sum_{i=1}^n |∑i=1n\textstyle\sum_{i=1}^n \textstyle\sum_{i=1}^n |x\scriptstyle x \scriptstyle x         (The size of a first sub/superscript) |x\scriptscriptstyle x \scriptscriptstyle x (The size of subsequent sub/superscripts) |lim⁡x\lim\limits_x \lim\limits_x |lim⁡x\lim\nolimits_x \lim\nolimits_x |x^2\verb!x^2! \verb!x^2!

\text{…} will accept nested $…$ fragments and render them in math mode.

12: Symbols and Punctuation ​

% comment…\dots \dotsKaTeX\KaTeX \KaTeX
%\% \%⋯\cdots \cdotsLaTeX\LaTeX \LaTeX
#\# \#⋱\ddots \ddotsTeX\TeX \TeX
&\& \&…\ldots \ldots∇\nabla \nabla
_\_ \_⋮\vdots \vdots∞\infty \infty
_\text{\textunderscore} \text{\textunderscore}⋯\dotsb \dotsb∞\infin \infin
–\text{--} \text{--}…\dotsc \dotsc✓\checkmark \checkmark
–\text{\textendash} \text{\textendash} ⁣⋯\dotsi \dotsi†\dag \dag
—\text{---} \text{---}⋯\dotsm \dotsm†\dagger \dagger
—\text{\textemdash} \text{\textemdash}…\dotso \dotso†\text{\textdagger} \text{\textdagger}
~\text{\textasciitilde} \text{\textasciitilde}⋅\sdot \sdot‡\ddag \ddag
‘` `…\mathellipsis \mathellipsis‡\ddagger \ddagger
‘\text{\textquoteleft} text{\textquoteleft}…\text{\textellipsis} \text{\textellipsis}‡\text{\textdaggerdbl} \text{\textdaggerdbl}
‘\lq \lq□\Box \Box‡\Dagger \Dagger
’\text{\textquoteright} \text{\textquoteright}□\square \square∠\angle \angle
′\rq \rq■\blacksquare \blacksquare∡\measuredangle \measuredangle
“\text{\textquotedblleft} \text{\textquotedblleft}△\triangle \triangle∢\sphericalangle \sphericalangle
"" "▽\triangledown \triangledown⊤\top \top
”\text{\textquotedblright} \text{\textquotedblright}◃\triangleleft \triangleleft⊥\bot \bot
 ⁣:\colon \colon▹\triangleright \triangleright$\$ \$
‵\backprime \backprime▽\bigtriangledown \bigtriangledown$\text{\textdollar} \text{\textdollar}
′\prime \prime△\bigtriangleup \bigtriangleup£\pounds \pounds
<\text{\textless} \text{\textless}▲\blacktriangle \blacktriangle£\mathsterling \mathsterling
>\text{\textgreater} \text{\textgreater}▼\blacktriangledown \blacktriangledown£\text{\textsterling} \text{\textsterling}
|\text{\textbar} \text{\textbar}◀\blacktriangleleft \blacktriangleleft¥\yen \yen
∥\text{\textbardbl} \text{\textbardbl}▶\blacktriangleright \blacktriangleright√\surd \surd
{\text{\textbraceleft} \text{\textbraceleft}⋄\diamond \diamond°\degree \degree
}\text{\textbraceright} \text{\textbraceright}◊\Diamond \Diamond°\text{\textdegree} \text{\textdegree}
\text{\P} \text{\P}◊\lozenge \lozenge℧\mho \mho
§\text{\S} \text{\S}⧫\blacklozenge \blacklozenge╲\diagdown \diagdown
§\text{\sect} \text{\sect}⋆\star \star╱\diagup \diagup
©\copyright \copyright★\bigstar \bigstar♭\flat \flat
®\circledR \circledR♣\clubsuit \clubsuit♮\natural \natural
®\text{\textregistered} \text{\textregistered}♣\clubs \clubs♯\sharp \sharp
Ⓢ\circledS \circledS♢\diamondsuit \diamondsuit♡\heartsuit \heartsuit
a◯\text{\textcircled a} \text{\textcircled a}♢\diamonds \diamonds♡\hearts \hearts
✠\maltese \maltese♠\spadesuit \spadesuit♠\spades \spades

Direct Input: £¥∇∞⋅∠∡∢♠♡♢♣♭♮♯✓…⋮⋯⋱!£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ ! ‼

13: Units ​

In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.

KaTeX UnitValueKaTeX UnitValue
emCSS embp1/72​ inch × F × G
exCSS expc12 KaTeX pt
mu1/18 CSS emdd1238/1157​ KaTeX pt
pt1/72.27 inch × F × Gcc14856/1157 KaTeX pt
mm1 mm × F × Gnd685/642 KaTeX pt
cm1 cm × F × Gnc1370/107​ KaTeX pt
in1 inch × F × Gsp1/65536 KaTeX pt

where:

The effect of style and size:

Unittextstylescriptscripthuge
em or ex\rule{1em}{1em}\scriptscriptstyle\rule{1em}{1em}\huge\rule{1em}{1em}
mu\rule{18mu}{18mu}\scriptscriptstyle\rule{18mu}{18mu}\huge\rule{18mu}{18mu}
others\rule{10pt}{10pt}\scriptscriptstyle\rule{10pt}{10pt}\huge\rule{10pt}{10pt}
- +M0 50 h400000 v40H0z m0 194h40000v40H0z"> \xlongequal{abc}

Extensible arrows all can take an optional argument in the same manner, as \xrightarrow[under]{over}.

11: Style, Color, Size, and Font ​

Class Assignment

Color

F=ma\color{blue} F=ma \color{blue} F=ma

Note that KaTeX \color acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:

For color definition, KaTeX color functions will accept the standard HTML predefined color names. They will also accept an RGB argument in CSS hexa­decimal style. The "#" is optional before a six-digit specification.

Font

Ab0\mathrm{Ab0} \mathrm{Ab0}Ab0\mathbf{Ab0} \mathbf{Ab0}Ab\mathit{Ab} \mathit{Ab}
Ab0\mathnormal{Ab0} \mathnormal{Ab0}Ab0\textbf{Ab0} \textbf{Ab0}Ab\textit{Ab} \textit{Ab}
Ab0\textrm{Ab0} \textrm{Ab0}Ab0\bf Ab0 \bf Ab0Ab\it Ab \it Ab
Ab0\rm Ab0 \rm Ab0Ab0\bold{Ab0} \bold{Ab0}AB\Bbb{AB} \Bbb{AB}
Ab0\textnormal{Ab0} \textnormal{Ab0}Ab\boldsymbol{Ab} \boldsymbol{Ab}AB\mathbb{AB} \mathbb{AB}
Ab0\text{Ab0} \text{Ab0}Ab\bm{Ab} \bm{Ab}Ab0\frak{Ab0} \frak{Ab0}
Ab0\mathsf{Ab0} \mathsf{Ab0}Ab0\mathtt{Ab0} \mathtt{Ab0}Ab0\mathfrak{Ab0} \mathfrak{Ab0}
Ab0\textsf{Ab0} \textsf{Ab0}Ab0\texttt{Ab0} \texttt{Ab0}AB0\mathcal{AB0} \mathcal{AB0}
Ab0\sf Ab0 \sf Ab0Ab0\tt Ab0 \tt Ab0AB\mathscr{AB} \mathscr{AB}

One can stack font family, font weight, and font shape by using the \textXX versions of the font functions. So \textsf{\textbf{H}} will produce H\textsf{\textbf{H}}. The other versions do not stack, e.g., \mathsf{\mathbf{H}} will produce H\mathsf{\mathbf{H}}.

In cases where KaTeX fonts do not have a bold glyph, \pmb can simulate one. For example, \pmb{\mu} renders as : μ\pmb{\mu}

Size

AB\Huge AB \Huge ABAB\normalsize AB \normalsize AB
AB\huge AB \huge ABAB\small AB \small AB
AB\LARGE AB \LARGE ABAB\footnotesize AB \footnotesize AB
AB\Large AB \Large ABAB\scriptsize AB \scriptsize AB
AB\large AB \large ABAB\tiny AB \tiny AB

Style

|∑i=1n\displaystyle\sum_{i=1}^n \displaystyle\sum_{i=1}^n |∑i=1n\textstyle\sum_{i=1}^n \textstyle\sum_{i=1}^n |x\scriptstyle x \scriptstyle x         (The size of a first sub/superscript) |x\scriptscriptstyle x \scriptscriptstyle x (The size of subsequent sub/superscripts) |lim⁡x\lim\limits_x \lim\limits_x |lim⁡x\lim\nolimits_x \lim\nolimits_x |x^2\verb!x^2! \verb!x^2!

\text{…} will accept nested $…$ fragments and render them in math mode.

12: Symbols and Punctuation ​

% comment…\dots \dotsKaTeX\KaTeX \KaTeX
%\% \%⋯\cdots \cdotsLaTeX\LaTeX \LaTeX
#\# \#⋱\ddots \ddotsTeX\TeX \TeX
&\& \&…\ldots \ldots∇\nabla \nabla
_\_ \_⋮\vdots \vdots∞\infty \infty
_\text{\textunderscore} \text{\textunderscore}⋯\dotsb \dotsb∞\infin \infin
–\text{--} \text{--}…\dotsc \dotsc✓\checkmark \checkmark
–\text{\textendash} \text{\textendash} ⁣⋯\dotsi \dotsi†\dag \dag
—\text{---} \text{---}⋯\dotsm \dotsm†\dagger \dagger
—\text{\textemdash} \text{\textemdash}…\dotso \dotso†\text{\textdagger} \text{\textdagger}
~\text{\textasciitilde} \text{\textasciitilde}⋅\sdot \sdot‡\ddag \ddag
‘` `…\mathellipsis \mathellipsis‡\ddagger \ddagger
‘\text{\textquoteleft} text{\textquoteleft}…\text{\textellipsis} \text{\textellipsis}‡\text{\textdaggerdbl} \text{\textdaggerdbl}
‘\lq \lq□\Box \Box‡\Dagger \Dagger
’\text{\textquoteright} \text{\textquoteright}□\square \square∠\angle \angle
′\rq \rq■\blacksquare \blacksquare∡\measuredangle \measuredangle
“\text{\textquotedblleft} \text{\textquotedblleft}△\triangle \triangle∢\sphericalangle \sphericalangle
"" "▽\triangledown \triangledown⊤\top \top
”\text{\textquotedblright} \text{\textquotedblright}◃\triangleleft \triangleleft⊥\bot \bot
 ⁣:\colon \colon▹\triangleright \triangleright$\$ \$
‵\backprime \backprime▽\bigtriangledown \bigtriangledown$\text{\textdollar} \text{\textdollar}
′\prime \prime△\bigtriangleup \bigtriangleup£\pounds \pounds
<\text{\textless} \text{\textless}▲\blacktriangle \blacktriangle£\mathsterling \mathsterling
>\text{\textgreater} \text{\textgreater}▼\blacktriangledown \blacktriangledown£\text{\textsterling} \text{\textsterling}
|\text{\textbar} \text{\textbar}◀\blacktriangleleft \blacktriangleleft¥\yen \yen
∥\text{\textbardbl} \text{\textbardbl}▶\blacktriangleright \blacktriangleright√\surd \surd
{\text{\textbraceleft} \text{\textbraceleft}⋄\diamond \diamond°\degree \degree
}\text{\textbraceright} \text{\textbraceright}◊\Diamond \Diamond°\text{\textdegree} \text{\textdegree}
\text{\P} \text{\P}◊\lozenge \lozenge℧\mho \mho
§\text{\S} \text{\S}⧫\blacklozenge \blacklozenge╲\diagdown \diagdown
§\text{\sect} \text{\sect}⋆\star \star╱\diagup \diagup
©\copyright \copyright★\bigstar \bigstar♭\flat \flat
®\circledR \circledR♣\clubsuit \clubsuit♮\natural \natural
®\text{\textregistered} \text{\textregistered}♣\clubs \clubs♯\sharp \sharp
Ⓢ\circledS \circledS♢\diamondsuit \diamondsuit♡\heartsuit \heartsuit
a◯\text{\textcircled a} \text{\textcircled a}♢\diamonds \diamonds♡\hearts \hearts
✠\maltese \maltese♠\spadesuit \spadesuit♠\spades \spades

Direct Input: £¥∇∞⋅∠∡∢♠♡♢♣♭♮♯✓…⋮⋯⋱!£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ ! ‼

13: Units ​

In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.

KaTeX UnitValueKaTeX UnitValue
emCSS embp1/72​ inch × F × G
exCSS expc12 KaTeX pt
mu1/18 CSS emdd1238/1157​ KaTeX pt
pt1/72.27 inch × F × Gcc14856/1157 KaTeX pt
mm1 mm × F × Gnd685/642 KaTeX pt
cm1 cm × F × Gnc1370/107​ KaTeX pt
in1 inch × F × Gsp1/65536 KaTeX pt

where:

The effect of style and size:

Unittextstylescriptscripthuge
em or ex\rule{1em}{1em}\scriptscriptstyle\rule{1em}{1em}\huge\rule{1em}{1em}
mu\rule{18mu}{18mu}\scriptscriptstyle\rule{18mu}{18mu}\huge\rule{18mu}{18mu}
others\rule{10pt}{10pt}\scriptscriptstyle\rule{10pt}{10pt}\huge\rule{10pt}{10pt}
+ \ No newline at end of file diff --git a/application/markdown-it-katex/support-table.html b/application/markdown-it-katex/support-table.html index 4b715d80..56c7f0f6 100644 --- a/application/markdown-it-katex/support-table.html +++ b/application/markdown-it-katex/support-table.html @@ -5,12 +5,12 @@ KaTeX: Support table | Toshiki's Note - + - + @@ -304,9 +304,9 @@ c-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333 -19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70 - 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z">\xtwoheadrightarrow{abc}

YZ ​

Symbol/FunctionRenderedSource or Comment
\yen¥\yen
\ZZ\Z
\ZetaZ\Zeta
\zetaζ\zeta
- + 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z">\xtwoheadrightarrow{abc}

YZ ​

Symbol/FunctionRenderedSource or Comment
\yen¥\yen
\ZZ\Z
\ZetaZ\Zeta
\zetaζ\zeta
+ \ No newline at end of file diff --git a/application/markdown-it-katex/tips.html b/application/markdown-it-katex/tips.html index 37e39b75..d84aa56c 100644 --- a/application/markdown-it-katex/tips.html +++ b/application/markdown-it-katex/tips.html @@ -5,12 +5,12 @@ Tips to Use markdown-it-katex | Toshiki's Note - + - + @@ -32,7 +32,7 @@ -
Toshiki's Note

Appearance

TOC

Tips & Hacks ​

1: Responsive KaTeX Styling ​

1.1: Issue background ​

KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.

katex overflowing on small screen devices
◎ katex overflowing on small screen devices

This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,

katex overflowing with labels annotated
◎ katex overflowing with labels annotated

The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.

1.2: Temporary solution ​

If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.

Break/wrap formula if too wide · Issue #327 · KaTeX/KaTeX
Is there a way to break formulaundefineds if they are to wide (with respect to their parent) ?
GitHub

But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.

1.3: Finding solution ​

After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.

GitHub - vuepress-theme-hope/vuepress-theme-hope: A vuepress theme with tons of features✨
A vuepress theme with tons of features✨. Contribute to vuepress-theme-hope/vuepress-theme-hope development by creating an account on GitHub.
GitHub

Under the styles directory of the repository, several line of scss styles came across my eyes,

KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme

this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,

Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen

Consequently, the fix is easy.

1.4: Solution ​

If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.

Install global scss support using your favored package manager.

sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss

Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.

scss
...
+    
Toshiki's Note
Appearance
TOC
Tips & Hacks ​
1: Responsive KaTeX Styling ​
1.1: Issue background ​
KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.
katex overflowing on small screen devices
◎ katex overflowing on small screen devices
This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,
katex overflowing with labels annotated
◎ katex overflowing with labels annotated
The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.
1.2: Temporary solution ​
If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.
Break/wrap formula if too wide · Issue #327 · KaTeX/KaTeX
Is there a way to break formulaundefineds if they are to wide (with respect to their parent) ?
GitHub
But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.
1.3: Finding solution ​
After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.
GitHub - vuepress-theme-hope/vuepress-theme-hope: A vuepress theme with tons of features✨
A vuepress theme with tons of features✨. Contribute to vuepress-theme-hope/vuepress-theme-hope development by creating an account on GitHub.
GitHub
Under the styles directory of the repository, several line of scss styles came across my eyes,
KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme
this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,
Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen
Consequently, the fix is easy.
1.4: Solution ​
If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.
Install global scss support using your favored package manager.
sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss
Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.
scss
...
 
 // katex responsiveness fix
 .katex {
@@ -128,9 +128,9 @@
 
 .katex-error {
     color: #f00;
-}
To be continued.
Reference

-    
+}

To be continued.

Reference

+ \ No newline at end of file diff --git a/assets/academic_chemistry_index.md.e22e1eca.js b/assets/academic_chemistry_index.md.20e560a6.js similarity index 90% rename from assets/academic_chemistry_index.md.e22e1eca.js rename to assets/academic_chemistry_index.md.20e560a6.js index f072b7b1..2144481f 100644 --- a/assets/academic_chemistry_index.md.e22e1eca.js +++ b/assets/academic_chemistry_index.md.20e560a6.js @@ -1 +1 @@ -import{_ as t,c as a,o as c,z as e,a as o}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1684858945000}'),s={name:"academic/chemistry/index.md"},r=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[o("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1),i=[r];function d(m,n,l,h,_,p){return c(),a("div",null,i)}const y=t(s,[["render",d]]);export{x as __pageData,y as default}; +import{_ as t,c as a,o as c,z as e,a as o}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1684859551000}'),s={name:"academic/chemistry/index.md"},r=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[o("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1),i=[r];function d(m,n,l,h,_,p){return c(),a("div",null,i)}const y=t(s,[["render",d]]);export{x as __pageData,y as default}; diff --git a/assets/academic_chemistry_index.md.e22e1eca.lean.js b/assets/academic_chemistry_index.md.20e560a6.lean.js similarity index 90% rename from assets/academic_chemistry_index.md.e22e1eca.lean.js rename to assets/academic_chemistry_index.md.20e560a6.lean.js index f072b7b1..2144481f 100644 --- a/assets/academic_chemistry_index.md.e22e1eca.lean.js +++ b/assets/academic_chemistry_index.md.20e560a6.lean.js @@ -1 +1 @@ -import{_ as t,c as a,o as c,z as e,a as o}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1684858945000}'),s={name:"academic/chemistry/index.md"},r=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[o("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1),i=[r];function d(m,n,l,h,_,p){return c(),a("div",null,i)}const y=t(s,[["render",d]]);export{x as __pageData,y as default}; +import{_ as t,c as a,o as c,z as e,a as o}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Chemistry","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/index.md","filePath":"academic/chemistry/index.md","lastUpdated":1684859551000}'),s={name:"academic/chemistry/index.md"},r=e("h1",{id:"welcome-to-chemistry",tabindex:"-1"},[o("Welcome to Chemistry "),e("a",{class:"header-anchor",href:"#welcome-to-chemistry","aria-label":'Permalink to "Welcome to Chemistry"'},"​")],-1),i=[r];function d(m,n,l,h,_,p){return c(),a("div",null,i)}const y=t(s,[["render",d]]);export{x as __pageData,y as default}; diff --git a/assets/academic_chemistry_notes_12-5.md.fa35f18c.js b/assets/academic_chemistry_notes_12-5.md.7fe55c03.js similarity index 99% rename from assets/academic_chemistry_notes_12-5.md.fa35f18c.js rename to assets/academic_chemistry_notes_12-5.md.7fe55c03.js index dfedcdc0..17f38d1c 100644 --- a/assets/academic_chemistry_notes_12-5.md.fa35f18c.js +++ b/assets/academic_chemistry_notes_12-5.md.7fe55c03.js @@ -1 +1 @@ -import{_ as t,c as l,o as e,R as n,z as s,a}from"./chunks/framework.a1bac067.js";const ls=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"tags":["Java","JVM"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1684858945000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n('

12-5: Reaction Mechanism ​

12-5-1: Learning Objectives ​

Learning Objectives

One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.

In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:

',4),r=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),p=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),h=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),o=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),u=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),v=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),w=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),x=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),b=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),k=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),f=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),z=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),O=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),_=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),M=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),N=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),C=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),L=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),A=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),T=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),B=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),R=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),q=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),I=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),S=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),P=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),D=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),E=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),F=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),H=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),V=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),j=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),U=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),$=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),J=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),W=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1),G=[i,r,c,p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,T,B,R,q,I,S,P,D,E,F,H,V,j,U,$,J,W];function Y(K,Q,X,Z,ss,as){return e(),l("div",null,G)}const es=t(m,[["render",Y]]);export{ls as __pageData,es as default}; +import{_ as t,c as l,o as e,R as n,z as s,a}from"./chunks/framework.a1bac067.js";const ls=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"tags":["Java","JVM"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1684859551000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n('

12-5: Reaction Mechanism ​

12-5-1: Learning Objectives ​

Learning Objectives

One of the major reasons for studying chemical kinetics is to use measurements of the macroscopic properties of a system, such as the rate of change in the concentration of reactants or products with time, to discover the sequence of events that occur at the molecular level during a reaction. This molecular description is the mechanism of the reaction; it describes how individual atoms, ions, or molecules interact to form particular products. The stepwise changes are collectively called the reaction mechanism.

In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:

',4),r=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),p=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),h=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),o=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),u=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),v=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),w=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),x=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),b=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),k=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),f=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),z=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),O=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),_=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),M=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),N=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),C=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),L=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),A=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),T=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),B=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),R=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),q=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),I=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),S=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),P=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),D=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),E=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),F=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),H=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),V=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),j=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),U=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),$=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),J=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),W=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1),G=[i,r,c,p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,T,B,R,q,I,S,P,D,E,F,H,V,j,U,$,J,W];function Y(K,Q,X,Z,ss,as){return e(),l("div",null,G)}const es=t(m,[["render",Y]]);export{ls as __pageData,es as default}; diff --git a/assets/academic_chemistry_notes_12-5.md.fa35f18c.lean.js b/assets/academic_chemistry_notes_12-5.md.7fe55c03.lean.js similarity index 99% rename from assets/academic_chemistry_notes_12-5.md.fa35f18c.lean.js rename to assets/academic_chemistry_notes_12-5.md.7fe55c03.lean.js index cfc6b5d2..c9b10d81 100644 --- a/assets/academic_chemistry_notes_12-5.md.fa35f18c.lean.js +++ b/assets/academic_chemistry_notes_12-5.md.7fe55c03.lean.js @@ -1 +1 @@ -import{_ as t,c as l,o as e,R as n,z as s,a}from"./chunks/framework.a1bac067.js";const ls=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"tags":["Java","JVM"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1684858945000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n("",4),r=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),p=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),h=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),o=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),u=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),v=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),w=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),x=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),b=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),k=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),f=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),z=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),O=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),_=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),M=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),N=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),C=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),L=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),A=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),T=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),B=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),R=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),q=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),I=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),S=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),P=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),D=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),E=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),F=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),H=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),V=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),j=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),U=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),$=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),J=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),W=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1),G=[i,r,c,p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,T,B,R,q,I,S,P,D,E,F,H,V,j,U,$,J,W];function Y(K,Q,X,Z,ss,as){return e(),l("div",null,G)}const es=t(m,[["render",Y]]);export{ls as __pageData,es as default}; +import{_ as t,c as l,o as e,R as n,z as s,a}from"./chunks/framework.a1bac067.js";const ls=JSON.parse('{"title":"12-5: Reaction Mechanism","description":"","frontmatter":{"title":"12-5: Reaction Mechanism","editLink":true,"lastUpdated":true,"showArticleMetadata":true,"categories":["Chemistry"],"tags":["Java","JVM"]},"headers":[],"relativePath":"academic/chemistry/notes/12-5.md","filePath":"academic/chemistry/notes/12-5.md","lastUpdated":1684859551000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n("",4),r=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"8")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"18")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"25"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("mn",null,"16"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"g")]),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"18"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"O"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{C}_{8} \\mathrm{H}_{18}(\\mathrm{l})+25 \\mathrm{O}_{2}(\\mathrm{~g}) \\longrightarrow 16 \\mathrm{CO}_{2}(\\mathrm{~g})+18 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"8")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"18")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"25"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"18"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathrm"},"O"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,"For this reaction to occur in a single step, 25 dioxygen molecules and 2 isooctane molecules would have to collide simultaneously and be converted to 34 molecules of product, which is very unlikely. It is more likely that a complex series of reactions takes place in a stepwise fashion. Each individual reaction, which is called an elementary reaction, involves one, two, or (rarely) three atoms, molecules, or ions. The overall sequence of elementary reactions is the mechanism of the reaction. The sum of the individual steps, or elementary reactions, in the mechanism must give the balanced chemical equation for the overall reaction.",-1),p=s("p",null,"The overall sequence of elementary reactions is the mechanism of the reaction.",-1),h=s("h2",{id:"_12-5-2-molecularity-and-the-rate-determining-step",tabindex:"-1"},[a("12-5-2: Molecularity and the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-2-molecularity-and-the-rate-determining-step","aria-label":'Permalink to "12-5-2: Molecularity and the Rate-Determining Step"'},"​")],-1),o=s("p",null,"To demonstrate how the analysis of elementary reactions helps us determine the overall reaction mechanism, we will examine the much simpler reaction of carbon monoxide with nitrogen dioxide.",-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O"),s("mo",null,"+"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"→"),s("mn",null,"2"),s("mi",{mathvariant:"normal"},"C"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{2CO + 2NO_2 \\rightarrow 2CO_2 + N_2O_4} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"2CO"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"2N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"2C"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"4")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])])])],-1),u=s("p",null,[a("From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with a molecule of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" that results in the transfer of an oxygen atom from nitrogen to carbon. The experimentally determined rate law for the reaction, however, is as follows:")],-1),d=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right]^{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])],-1),v=s("p",null,[a("The fact that the reaction is second order in "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\left[\\mathrm{NO}_{2}\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" and independent of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(" tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be")],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_{2}\\right][\\mathrm{CO}] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),w=s("p",null,"The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:",-1),x=s("h3",{id:"_15-2-1-two-step-mechanism",tabindex:"-1"},[a("15-2-1: Two-Step Mechanism "),s("a",{class:"header-anchor",href:"#_15-2-1-two-step-mechanism","aria-label":'Permalink to "15-2-1: Two-Step Mechanism"'},"​")],-1),b=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Reaction Type")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"⟶")]),s("mtext",null," slow ")])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{NO}_{2} \\stackrel{\\text { slow }}{\\longrightarrow} \\mathrm{NO}_{3}+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.4471em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2971em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"⟶")])]),s("span",{style:{top:"-3.711em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," slow ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.011em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_{3}+\\mathrm{CO} \\rightarrow \\mathrm{NO}_{2}+\\mathrm{CO}_{2}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0333em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.69em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,"elementary reaction")]),s("tr",null,[s("td",null,"sum"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"→"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}+\\mathrm{CO} \\rightarrow \\mathrm{NO}+\\mathrm{CO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,"overall reaction")])])],-1),k=s("p",null,[a("According to this mechanism, the overall reaction occurs in two steps, or elementary reactions. Summing steps 1 and 2 and canceling on both sides of the equation gives the overall balanced chemical equation for the reaction. The "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO_3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),a(" molecule is intermediate in the reaction, a species that does not appear in the balanced chemical equation for the overall reaction. It is formed as a product of the first step but is consumed in the second step.")],-1),f=s("p",null,"The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.",-1),z=s("h2",{id:"_12-5-3-using-molecularity-to-describe-a-rate-law",tabindex:"-1"},[a("12-5-3: Using Molecularity to Describe a Rate Law "),s("a",{class:"header-anchor",href:"#_12-5-3-using-molecularity-to-describe-a-rate-law","aria-label":'Permalink to "12-5-3: Using Molecularity to Describe a Rate Law"'},"​")],-1),O=s("p",null,"The molecularity of an elementary reaction is the number of molecules that collide during that step in the mechanism. If there is only a single reactant molecule in an elementary reaction, that step is designated as unimolecular; if there are two reactant molecules, it is bimolecular; and if there are three reactant molecules (a relatively rare situation), it is termolecular. Elementary reactions that involve the simultaneous collision of more than three molecules are highly improbable and have never been observed experimentally. (To understand why, try to make three or more marbles or pool balls collide with one another simultaneously!)",-1),_=s("div",{class:"warning custom-block"},[s("p",{class:"custom-block-title"},"About the image"),s("p",null,"The Basis for Writing Rate Laws of Elementary Reactions. This diagram illustrates how the number of possible collisions per unit time between two reactant species, A and B, depends on the number of A and B particles present. The number of collisions between A and B particles increases as the product of the number of particles, not as the sum. This is why the rate law for an elementary reaction depends on the product of the concentrations of the species that collide in that step. (CC BY-NC-SA; anonymous)")],-1),M=s("p",null,[a("Writing the rate law for an elementary reaction is straightforward because we know how many molecules must collide simultaneously for the elementary reaction to occur; hence the order of the elementary reaction is the same as its molecularity (Table 14.6.1). In contrast, the rate law for the reaction cannot be determined from the balanced chemical equation for the overall reaction. The general rate law for a unimolecular elementary reaction ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products) is")],-1),N=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),C=s("p",null,[a("For bimolecular reactions, the reaction rate depends on the number of collisions per unit time, which is proportional to the product of the concentrations of the reactants, as shown in Figure 14.6.1 For a bimolecular elementary reaction of the form "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→")])])]),a(" products, the general rate law is")],-1),L=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mi",null,"A"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",null,"B"),s("mo",{stretchy:"false"},"]"),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[A][B] . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mord"},".")])])])])],-1),A=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Elementary Reaction"),s("th",null,"Molecularity"),s("th",null,"Rate"),s("th",null,"Reaction Order")])]),s("tbody",null,[s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Unimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"first")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2\\mathrm{~A} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Bimolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"second")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~A}+\\mathrm{B} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}]^2[\\mathrm{~B}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")]),s("tr",null,[s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"A"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"B"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"→"),s("mtext",null," products ")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{A}+\\mathrm{B}+\\mathrm{C} \\rightarrow \\text { products }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," products ")])])])])]),s("td",null,"Termolecular"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k[\\mathrm{~A}][\\mathrm{B}][\\mathrm{C}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"B"),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"["),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mclose"},"]")])])])]),s("td",null,"third")])])],-1),T=s("blockquote",null,[s("p",null,"For elementary reactions, the order of the elementary reaction is the same as its molecularity. In contrast, the rate law cannot be determined from the balanced chemical equation for the overall reaction (unless it is a single step mechanism and is therefore also an elementary step).")],-1),B=s("h2",{id:"_12-5-4-identifying-the-rate-determining-step",tabindex:"-1"},[a("12-5-4: Identifying the Rate-Determining Step "),s("a",{class:"header-anchor",href:"#_12-5-4-identifying-the-rate-determining-step","aria-label":'Permalink to "12-5-4: Identifying the Rate-Determining Step"'},"​")],-1),R=s("p",null,"Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the ratedetermining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.",-1),q=s("p",null,"Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.",-1),I=s("p",null,"Rate laws for each elementary reaction in our example as well as for the overall reaction",-1),S=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Steps"),s("th",null,"Reaction"),s("th",null,"Rate")])]),s("tbody",null,[s("tr",null,[s("td",null,"step 1"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",{mathvariant:"normal"},"k"),s("mn",null,"1")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{NO}_2 \\stackrel{\\mathrm{k}_1}{\\rightarrow} \\mathrm{NO}_3+\\mathrm{NO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3141em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_1\\left[\\mathrm{NO}_2\\right]^2(\\text { predicted) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted) ")])])])])])]),s("tr",null,[s("td",null,"step 2"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("munder",{accentunder:"true"},[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")])])]),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("mo",{stretchy:"true"},"‾")])]),s("annotation",{encoding:"application/x-tex"},"\\underline{\\mathrm{NO}_3+\\mathrm{CO} \\stackrel{k_2}{\\rightarrow} \\mathrm{NO}_2+\\mathrm{CO}_2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.5141em","vertical-align":"-0.35em"}}),s("span",{class:"mord underline"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-2.8541em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"underline-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.1641em"}},[s("span",{class:"pstrut",style:{height:"3.1641em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1641em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.578em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.35em"}},[s("span")])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]"),s("mo",{stretchy:"false"},"("),s("mtext",null," predicted "),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k_2\\left[\\mathrm{NO}_3\\right][\\mathrm{CO}](\\text { predicted })")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]"),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," predicted ")]),s("span",{class:"mclose"},")")])])])])]),s("tr",null,[s("td",null,"step 3"),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",null,"+"),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,[s("mover",null,[s("mo",null,[s("mo",null,"→")]),s("mi",null,"k")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mo",null,"+"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_2+\\mathrm{CO} \\stackrel{k}{\\rightarrow} \\mathrm{NO}+\\mathrm{CO}_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.153em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.153em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",null,[s("span",{class:"mop"},"→")])]),s("span",{style:{top:"-3.5669em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])])]),s("td",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mi",null,"k"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mtext",null," observed) ")]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=k\\left[\\mathrm{NO}_2\\right]^2(\\text { observed) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord text"},[s("span",{class:"mord"}," observed) ")])])])])])])])],-1),P=s("p",null,[a("The experimentally determined rate law for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mi",null,"O")]),s("annotation",{encoding:"application/x-tex"},"C O")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"CO")])])]),a(" is the same as the predicted rate law for step 1 . This tells us that the first elementary reaction is the rate-determining step, so "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" for the overall reaction must equal "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". That is, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.")],-1),D=s("p",null,"Sometimes chemists are able to propose two or more mechanisms that are consistent with the available data. If a proposed mechanism predicts the wrong experimental rate law, however, the mechanism must be incorrect.",-1),E=s("h3",{id:"_12-5-4-1-example-a-reaction-with-an-intermediate",tabindex:"-1"},[a("12-5-4-1: Example-A Reaction with an Intermediate "),s("a",{class:"header-anchor",href:"#_12-5-4-1-example-a-reaction-with-an-intermediate","aria-label":'Permalink to "12-5-4-1: Example-A Reaction with an Intermediate"'},"​")],-1),F=s("p",null,[a("In an alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),H=s("p",null,[a("alternative mechanism for the reaction of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{CO}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")])])])]),a(" with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" appearing as an intermediate.")],-1),V=s("p",null,[a("Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mo",{stretchy:"false"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"t")])]),s("annotation",{encoding:"application/x-tex"},"=\\mathrm{k[{NO}_{2}]^{2}t}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])])])])])])]),s("span",{class:"mord mathrm"},"t")])])])]),a(")")],-1),j=s("p",null,"Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law",-1),U=s("h4",{id:"strategy",tabindex:"-1"},[a("Strategy "),s("a",{class:"header-anchor",href:"#strategy","aria-label":'Permalink to "Strategy"'},"​")],-1),$=s("ul",null,[s("li",null,[s("p",null,"Determine the rate law for each elementary reaction in the reaction.")]),s("li",null,[s("p",null,"Determine which rate law corresponds to the experimentally determined rate law for the reaction. This rate law is the one for the rate-determining step.")])],-1),J=s("h4",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),W=s("details",{class:"details custom-block"},[s("summary",null,"View solution"),s("p",null,[a("A The rate law for step 1 is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a("; for step 2 , it is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("msub",null,[s("mi",null,"k"),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")]),s("mo",{fence:"true"},"]")]),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"=k_{2}\\left[\\mathrm{~N}_{2} \\mathrm{O}_{4}\\right][\\mathrm{CO}]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CO")]),s("span",{class:"mclose"},"]")])])]),a(".")]),s("p",null,[a("B If step 1 is slow (and therefore the rate-determining step), then the overall rate law for the reaction will be the same: rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mn",null,"1")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"k_{1}\\left[\\mathrm{NO}_{2}\\right]^{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.204em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(". This is the same as the experimentally determined rate law. Hence this mechanism, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, and the one described previously, with "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as an intermediate, are kinetically indistinguishable. In this case, further experiments are needed to distinguish between them. For example, the researcher could try to detect the proposed intermediates, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{NO}_{3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"4")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_{2} \\mathrm{O}_{4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", directly.")])],-1),G=[i,r,c,p,h,o,g,u,d,v,y,w,x,b,k,f,z,O,_,M,N,C,L,A,T,B,R,q,I,S,P,D,E,F,H,V,j,U,$,J,W];function Y(K,Q,X,Z,ss,as){return e(),l("div",null,G)}const es=t(m,[["render",Y]]);export{ls as __pageData,es as default}; diff --git a/assets/academic_chemistry_problems_02-20.md.e1fa1931.js b/assets/academic_chemistry_problems_02-20.md.6c822dc9.js similarity index 99% rename from assets/academic_chemistry_problems_02-20.md.e1fa1931.js rename to assets/academic_chemistry_problems_02-20.md.6c822dc9.js index 0f80e352..0defa44b 100644 --- a/assets/academic_chemistry_problems_02-20.md.e1fa1931.js +++ b/assets/academic_chemistry_problems_02-20.md.6c822dc9.js @@ -1 +1 @@ -import{_ as t,c as l,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const O=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1684858945000}'),n={name:"academic/chemistry/problems/02-20.md"},e=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),p=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),h=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),o=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),g=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),d=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1),y=[e,r,i,p,c,h,o,g,u,d,v];function w(b,z,x,f,H,k){return m(),l("div",null,y)}const C=t(n,[["render",w]]);export{O as __pageData,C as default}; +import{_ as t,c as l,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const O=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1684859551000}'),n={name:"academic/chemistry/problems/02-20.md"},e=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),p=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),h=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),o=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),g=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),d=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1),y=[e,r,i,p,c,h,o,g,u,d,v];function w(b,z,x,f,H,k){return m(),l("div",null,y)}const C=t(n,[["render",w]]);export{O as __pageData,C as default}; diff --git a/assets/academic_chemistry_problems_02-20.md.e1fa1931.lean.js b/assets/academic_chemistry_problems_02-20.md.6c822dc9.lean.js similarity index 99% rename from assets/academic_chemistry_problems_02-20.md.e1fa1931.lean.js rename to assets/academic_chemistry_problems_02-20.md.6c822dc9.lean.js index 0f80e352..0defa44b 100644 --- a/assets/academic_chemistry_problems_02-20.md.e1fa1931.lean.js +++ b/assets/academic_chemistry_problems_02-20.md.6c822dc9.lean.js @@ -1 +1 @@ -import{_ as t,c as l,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const O=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1684858945000}'),n={name:"academic/chemistry/problems/02-20.md"},e=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),p=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),h=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),o=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),g=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),d=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1),y=[e,r,i,p,c,h,o,g,u,d,v];function w(b,z,x,f,H,k){return m(),l("div",null,y)}const C=t(n,[["render",w]]);export{O as __pageData,C as default}; +import{_ as t,c as l,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const O=JSON.parse('{"title":"Presentation problem: 02-20","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/02-20.md","filePath":"academic/chemistry/problems/02-20.md","lastUpdated":1684859551000}'),n={name:"academic/chemistry/problems/02-20.md"},e=s("h1",{id:"presentation-problem-02-20",tabindex:"-1"},[a("Presentation problem: 02-20 "),s("a",{class:"header-anchor",href:"#presentation-problem-02-20","aria-label":'Permalink to "Presentation problem: 02-20"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("At "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"K")])]),s("annotation",{encoding:"application/x-tex"},"500 \\mathrm{~K}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"K")])])])]),a(" in the presence of a copper surface, ethanol decomposes according to the equation")],-1),p=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"⟶"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H")]),s("mn",null,"3")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"C"),s("mi",{mathvariant:"normal"},"H"),s("mi",{mathvariant:"normal"},"O")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{g}) \\longrightarrow \\mathrm{CH}_3 \\mathrm{CHO}(g)+\\mathrm{H}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⟶"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CH")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"CHO")]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),c=s("p",null,[a("The pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" was measured as a function of time and the following data were obtained:")],-1),h=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])])]),s("mtext",null," (torr) ")]),s("annotation",{encoding:"application/x-tex"},"P_{\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}} \\text { (torr) }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0001em","vertical-align":"-0.2501em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"OH")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2501em"}},[s("span")])])])])]),s("span",{class:"mord text"},[s("span",{class:"mord"}," (torr) ")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"250.")]),s("tr",null,[s("td",null,"100."),s("td",null,"237")]),s("tr",null,[s("td",null,"200."),s("td",null,"224")]),s("tr",null,[s("td",null,"300."),s("td",null,"211")]),s("tr",null,[s("td",null,"400."),s("td",null,"198")]),s("tr",null,[s("td",null,"500."),s("td",null,"185")])])],-1),o=s("p",null,[s("strong",null,"Since the pressure of a gas is directly proportional to the concentration of gas"),a(", we can express the rate law for a gaseous reaction in terms of partial pressures. Using the above data, deduce the rate law, the integrated rate law, and the value of the rate constant, all in terms of pressure units in atm and time in seconds. Predict the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" after "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"900."),s("mi",{mathvariant:"normal"},"s")]),s("annotation",{encoding:"application/x-tex"},"900 . \\mathrm{s}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"900."),s("span",{class:"mord mathrm"},"s")])])]),a(" from the start of the reaction. (Hint: To determine the order of the reaction with respect to "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(", compare how the pressure of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")])])])]),a(" decreases with each time listing.)")],-1),g=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),u=s("p",null,[s("figure",null,[s("img",{src:"https://r2.toshiki.dev/image/chemistry/solution1.png",alt:"solution graph",title:""}),s("figcaption",{align:"center"},[s("small",null,"◎ ")])])],-1),d=s("p",null,[a("Due to the fact that the graph of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{fence:"true"},"]")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{p}\\left[\\mathrm{O}_2\\right]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),a(" over time is showing "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"R"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{R}^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(" value of 1 we know we have a zero-order reaction. Therefore:")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")]),s("mo",null,"−"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},"]")])]),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"250"),s("mtext",null," torr "),s("mo",null,"−"),s("mn",null,"185"),s("mtext",null," torr ")]),s("mrow",null,[s("mn",null,"500"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")])])]),s("mo",null,"="),s("mn",null,"0.13"),s("msup",null,[s("mtext",null," torr s "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"k"),s("mo",null,"="),s("mn",null,"0.13"),s("mtext",null," torr s "),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"×"),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])]),s("mrow",null,[s("mn",null,"760"),s("mtext",null," torr ")])]),s("mo",null,"="),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," rate "),s("mo",null,"="),s("mi",{mathvariant:"bold"},"k")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mi",{mathvariant:"normal"},"k"),s("mi",{mathvariant:"normal"},"t")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"p"),s("mn",null,"0")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"−"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"1.71"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{fence:"true"},")")]),s("mo",null,"×"),s("mn",null,"900"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"+"),s("mn",null,"0.33"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"normal"},"p"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"normal"},"C"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"H"),s("mn",null,"5")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"O"),s("mi",{mathvariant:"normal"},"H")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mn",null,"0.176"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"a"),s("mi",{mathvariant:"normal"},"t"),s("mi",{mathvariant:"normal"},"m")]),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\mathrm{k}=\\frac{\\mathrm{p}_0\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]-\\mathrm{p}\\left[\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right]}{\\mathrm{t}}=\\frac{250 \\text { torr }-185 \\text { torr }}{500 \\mathrm{~s}}=0.13 \\text { torr s }^{-1} \\\\ & \\mathrm{k}=0.13 \\text { torr s } \\mathrm{s}^{-1} \\times \\frac{1 \\mathrm{~atm}}{760 \\text { torr }}=1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\text { rate }=\\mathbf{k} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\mathrm{kt}+\\mathrm{p}_0\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right) \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=-\\left(1.71 \\times 10^{-4} \\mathrm{~atm} \\mathrm{~s}^{-1}\\right) \\times 900 \\mathrm{~s}+0.33 \\mathrm{~atm} \\\\ & \\mathrm{p}\\left(\\mathrm{C}_2 \\mathrm{H}_5 \\mathrm{OH}\\right)=0.176 \\mathrm{~atm} \\mathrm{~s}^{-1} \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"12.2687em","vertical-align":"-5.8843em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.3843em"}},[s("span",{style:{top:"-8.3843em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"500"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"250"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"185"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-6.0769em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.13"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr s ")]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"760"),s("span",{class:"mord text"},[s("span",{class:"mord"}," torr ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"-4.2509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathbf"},"k")])]),s("span",{style:{top:"-2.7509em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"kt")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-1.2268em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"1.71"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"900"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.33"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")])])]),s("span",{style:{top:"0.2973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathrm"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"C"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"H"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"OH")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.176"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"atm")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])]),s("span",{style:{top:"1.7973em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8843em"}},[s("span")])])])])])])])])])])],-1),y=[e,r,i,p,c,h,o,g,u,d,v];function w(b,z,x,f,H,k){return m(),l("div",null,y)}const C=t(n,[["render",w]]);export{O as __pageData,C as default}; diff --git a/assets/academic_chemistry_problems_03-02-1.md.3f33f487.js b/assets/academic_chemistry_problems_03-02-1.md.765a5cd1.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-1.md.3f33f487.js rename to assets/academic_chemistry_problems_03-02-1.md.765a5cd1.js index 83d458bf..593e857b 100644 --- a/assets/academic_chemistry_problems_03-02-1.md.3f33f487.js +++ b/assets/academic_chemistry_problems_03-02-1.md.765a5cd1.js @@ -1 +1 @@ -import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1684858945000}'),e={name:"academic/chemistry/problems/03-02-1.md"},m=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),p=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),r=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),o=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),h=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),u=s("p",null,"the rate of reaction is defined as",-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),x=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),f=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1),k=[m,c,i,p,r,o,h,g,u,y,d,v,b,x,w,f];function z(M,_,N,O,L,D){return n(),t("div",null,k)}const P=l(e,[["render",z]]);export{A as __pageData,P as default}; +import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1684859551000}'),e={name:"academic/chemistry/problems/03-02-1.md"},m=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),p=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),r=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),o=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),h=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),u=s("p",null,"the rate of reaction is defined as",-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),x=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),f=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1),k=[m,c,i,p,r,o,h,g,u,y,d,v,b,x,w,f];function z(M,_,N,O,L,D){return n(),t("div",null,k)}const P=l(e,[["render",z]]);export{A as __pageData,P as default}; diff --git a/assets/academic_chemistry_problems_03-02-1.md.3f33f487.lean.js b/assets/academic_chemistry_problems_03-02-1.md.765a5cd1.lean.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-1.md.3f33f487.lean.js rename to assets/academic_chemistry_problems_03-02-1.md.765a5cd1.lean.js index 83d458bf..593e857b 100644 --- a/assets/academic_chemistry_problems_03-02-1.md.3f33f487.lean.js +++ b/assets/academic_chemistry_problems_03-02-1.md.765a5cd1.lean.js @@ -1 +1 @@ -import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1684858945000}'),e={name:"academic/chemistry/problems/03-02-1.md"},m=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),p=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),r=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),o=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),h=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),u=s("p",null,"the rate of reaction is defined as",-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),x=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),f=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1),k=[m,c,i,p,r,o,h,g,u,y,d,v,b,x,w,f];function z(M,_,N,O,L,D){return n(),t("div",null,k)}const P=l(e,[["render",z]]);export{A as __pageData,P as default}; +import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Problem: 03-02-1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-1.md","filePath":"academic/chemistry/problems/03-02-1.md","lastUpdated":1684859551000}'),e={name:"academic/chemistry/problems/03-02-1.md"},m=s("h1",{id:"problem-03-02-1",tabindex:"-1"},[a("Problem: 03-02-1 "),s("a",{class:"header-anchor",href:"#problem-03-02-1","aria-label":'Permalink to "Problem: 03-02-1"'},"​")],-1),c=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"N")]),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"→"),s("mn",null,"4"),s("msub",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"N"),s("mi",{mathvariant:"normal"},"O")]),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2 \\mathrm{~N}_2 \\mathrm{O}_5(g) \\rightarrow 4 \\mathrm{NO}_2(g)+\\mathrm{O}_2(g) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"N")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"NO")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])])])],-1),p=s("p",null,[a("The decomposition of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is represented by the equation above. A sample of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"("),s("mi",null,"g"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5(g)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")")])])]),a(" is monitored as it decomposes, and the concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" as a function of time is recorded. The results are shown in the table below.")],-1),r=s("table",null,[s("thead",null,[s("tr",null,[s("th",null,"Time (s)"),s("th",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{[N_2O_5]}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathrm mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose"},"]")])])])])])])]),s("tbody",null,[s("tr",null,[s("td",null,"0"),s("td",null,"1.000")]),s("tr",null,[s("td",null,"25.0"),s("td",null,"0.801")]),s("tr",null,[s("td",null,"50.0"),s("td",null,"0.642")]),s("tr",null,[s("td",null,"75.0"),s("td",null,"0.515")])])],-1),o=s("p",null,[a("Calculate the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds.")],-1),h=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),g=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"A")]),s("mo",null,"+"),s("mi",null,"b"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"B")]),s("mo",null,"→"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"C"),s("mo",null,"+"),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"D")]),s("annotation",{encoding:"application/x-tex"},"a \\mathrm{~A}+b \\mathrm{~B} \\rightarrow c \\mathrm{C}+d \\mathrm{D} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"A")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"B")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"→"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mord mathrm"},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord mathrm"},"D")])])])])],-1),u=s("p",null,"the rate of reaction is defined as",-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mstyle",{mathcolor:"cyan"},[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"a")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"A"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"b")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"B"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"c")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"C"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])]),s("mo",null,"="),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mi",null,"d")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"D"),s("mo",{stretchy:"false"},"]")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])])]),s("annotation",{encoding:"application/x-tex"},"\\color{cyan}\\text { rate }=-\\frac{1}{a} \\frac{\\Delta[\\mathrm{A}]}{\\Delta t}=-\\frac{1}{b} \\frac{\\Delta[\\mathrm{B}]}{\\Delta t}=+\\frac{1}{c} \\frac{\\Delta[\\mathrm{C}]}{\\Delta t}=+\\frac{1}{d} \\frac{\\Delta[\\mathrm{D}]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"a")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"A"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"−"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"b")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"B"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"c")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"C"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel",style:{color:"cyan"}},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord",style:{color:"cyan"}},"+"),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord mathnormal",style:{color:"cyan"}},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mord mathnormal",style:{color:"cyan"}},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{color:"cyan","border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{color:"cyan"}},[s("span",{class:"mord",style:{color:"cyan"}},"Δ"),s("span",{class:"mopen",style:{color:"cyan"}},"["),s("span",{class:"mord mathrm",style:{color:"cyan"}},"D"),s("span",{class:"mclose",style:{color:"cyan"}},"]")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,[a("Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(", "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(", or "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" ). This ensures that the calculated reaction rate is the same no matter which reactant or product is monitored for changes in concentration. In this case, the monitored species was "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". With that in mind, let's write the reaction rate in terms of the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" :")],-1),v=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mtext",null," rate "),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mrow",null,[s("mo",{fence:"true"},"["),s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")]),s("mo",{fence:"true"},"]")])]),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t")])])]),s("annotation",{encoding:"application/x-tex"},"\\text { rate }=-\\frac{1}{2} \\frac{\\Delta\\left[\\mathrm{N}_2 \\mathrm{O}_5\\right]}{\\Delta t} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.113em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"Δ"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"]")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("p",null,[a("Since the coefficient for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" in the balanced equation is 2 , we divided the rate of change in concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" by 2 . Additionally, since "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"normal"},"N"),s("mn",null,"2")]),s("msub",null,[s("mi",{mathvariant:"normal"},"O"),s("mn",null,"5")])]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{N}_2 \\mathrm{O}_5")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"N"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"O"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"5")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" is being consumed in the reaction, we included a negative sign in front of the expression.")],-1),x=s("p",null,[a("Now, let's plug in the information from the table to calculate the average reaction rate between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds:")],-1),w=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mtext",null," rate ")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mo",null,"−"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.515"),s("mi",null,"M"),s("mo",null,"−"),s("mn",null,"0.642"),s("mi",null,"M"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"75.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",null,"−"),s("mn",null,"50.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mo",{stretchy:"false"},")")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"2.54"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])]),s("mi",null,"M"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"s")]),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\text { rate } & =-\\frac{1}{2} \\frac{(0.515 M-0.642 M)}{(75.0 \\mathrm{~s}-50.0 \\mathrm{~s})} \\\\ & =2.54 \\times 10^{-3} M \\mathrm{~s}^{-1} \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.1871em","vertical-align":"-1.8436em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3436em"}},[s("span",{style:{top:"-4.3436em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"75.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"50.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.515"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"0.642"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-2.2434em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"s")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.8436em"}},[s("span")])])])])])])])])])])],-1),f=s("p",null,[a("So, the average rate of the reaction between "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"50.0")]),s("annotation",{encoding:"application/x-tex"},"50.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"50.0")])])]),a(" and "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"75.0")]),s("annotation",{encoding:"application/x-tex"},"75.0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"75.0")])])]),a(" seconds is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",{mathvariant:"bold"},"2.54"),s("mo",null,"×"),s("mn",{mathvariant:"bold"},"1"),s("msup",null,[s("mn",{mathvariant:"bold"},"0"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"3")])]),s("mi",{mathvariant:"normal"},"M"),s("msup",null,[s("mi",{mathvariant:"normal"},"s"),s("mrow",null,[s("mo",null,"−"),s("mn",{mathvariant:"bold"},"1")])])]),s("annotation",{encoding:"application/x-tex"},"\\bold{2.54 \\times 10^{-3} \\mathrm{M} \\mathrm{s}^{-1}}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"2.54"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathbf"},"1"),s("span",{class:"mord"},[s("span",{class:"mord mathbf"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"3")])])])])])])])]),s("span",{class:"mord mathrm"},"M"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"s"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathbf mtight"},"1")])])])])])])])])])])])]),a(".")],-1),k=[m,c,i,p,r,o,h,g,u,y,d,v,b,x,w,f];function z(M,_,N,O,L,D){return n(),t("div",null,k)}const P=l(e,[["render",z]]);export{A as __pageData,P as default}; diff --git a/assets/academic_chemistry_problems_03-02-2.md.bab6348d.js b/assets/academic_chemistry_problems_03-02-2.md.35916f88.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-2.md.bab6348d.js rename to assets/academic_chemistry_problems_03-02-2.md.35916f88.js index f925be94..966c3bc8 100644 --- a/assets/academic_chemistry_problems_03-02-2.md.bab6348d.js +++ b/assets/academic_chemistry_problems_03-02-2.md.35916f88.js @@ -1 +1 @@ -import{_ as l,c as t,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1684858945000}'),n={name:"academic/chemistry/problems/03-02-2.md"},e=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),c=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),p=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),o=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),u=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),g=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1),y=[e,r,i,c,p,o,h,d,u,g];function v(x,w,k,b,f,L){return m(),t("div",null,y)}const z=l(n,[["render",v]]);export{_ as __pageData,z as default}; +import{_ as l,c as t,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1684859551000}'),n={name:"academic/chemistry/problems/03-02-2.md"},e=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),c=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),p=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),o=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),u=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),g=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1),y=[e,r,i,c,p,o,h,d,u,g];function v(x,w,k,b,f,L){return m(),t("div",null,y)}const z=l(n,[["render",v]]);export{_ as __pageData,z as default}; diff --git a/assets/academic_chemistry_problems_03-02-2.md.bab6348d.lean.js b/assets/academic_chemistry_problems_03-02-2.md.35916f88.lean.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-2.md.bab6348d.lean.js rename to assets/academic_chemistry_problems_03-02-2.md.35916f88.lean.js index f925be94..966c3bc8 100644 --- a/assets/academic_chemistry_problems_03-02-2.md.bab6348d.lean.js +++ b/assets/academic_chemistry_problems_03-02-2.md.35916f88.lean.js @@ -1 +1 @@ -import{_ as l,c as t,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1684858945000}'),n={name:"academic/chemistry/problems/03-02-2.md"},e=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),c=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),p=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),o=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),u=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),g=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1),y=[e,r,i,c,p,o,h,d,u,g];function v(x,w,k,b,f,L){return m(),t("div",null,y)}const z=l(n,[["render",v]]);export{_ as __pageData,z as default}; +import{_ as l,c as t,o as m,z as s,a}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem: 03-02-2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-2.md","filePath":"academic/chemistry/problems/03-02-2.md","lastUpdated":1684859551000}'),n={name:"academic/chemistry/problems/03-02-2.md"},e=s("h1",{id:"problem-03-02-2",tabindex:"-1"},[a("Problem: 03-02-2 "),s("a",{class:"header-anchor",href:"#problem-03-02-2","aria-label":'Permalink to "Problem: 03-02-2"'},"​")],-1),r=s("h2",{id:"question",tabindex:"-1"},[a("Question "),s("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=s("p",null,[a("The rate law for a particular reaction is rate "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mi",null,"k"),s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{XY}]^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(". In an experiment, the initial rate of the reaction is determined to be "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),a(" when the initial concentration of "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{XY}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")])])])]),a(" is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~mol} / \\mathrm{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L")])])]),a(".")],-1),c=s("ul",null,[s("li",null,[a("What is the value of the rate constant, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(", for the reaction? "),s("ul",{class:"contains-task-list"},[s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.10"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.10 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.10"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"0.40 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",checked:"",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])]),s("li",{class:"task-list-item"},[s("input",{class:"task-list-item-checkbox",disabled:"",type:"checkbox"}),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.5"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"2.5 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2.5"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])])])])],-1),p=s("h2",{id:"solution",tabindex:"-1"},[a("Solution "),s("a",{class:"header-anchor",href:"#solution","aria-label":'Permalink to "Solution"'},"​")],-1),o=s("p",null,[a("lo tind the value of the rate constant for the reaction, let's first solve the rate law for "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k")]),s("annotation",{encoding:"application/x-tex"},"k")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])])]),a(" :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"k"),s("mo",null,"="),s("mfrac",null,[s("mtext",null," rate "),s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mrow",null,[s("mi",{mathvariant:"normal"},"X"),s("mi",{mathvariant:"normal"},"Y")]),s("msup",null,[s("mo",{stretchy:"false"},"]"),s("mn",null,"2")])])])]),s("annotation",{encoding:"application/x-tex"},"k=\\frac{\\text { rate }}{[\\mathrm{XY}]^2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2281em","vertical-align":"-0.936em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2921em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"XY")]),s("span",{class:"mclose"},[s("span",{class:"mclose"},"]"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," rate ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),d=s("p",null,"Next, let's plug in the initial rate and concentration given in the text:",-1),u=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"k")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mn",null,"0.40"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"L"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"0.16"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"L"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mn",null,"0.16"),s("msup",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",{mathvariant:"normal"},"L"),s("mn",null,"2")])])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} k & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{(0.40 \\mathrm{~mol} / \\mathrm{L})^2} \\\\ & =\\frac{0.16 \\mathrm{~mol} /(\\mathrm{L} \\cdot \\mathrm{s})}{0.16 \\mathrm{~mol}^2 / \\mathrm{L}^2} \\\\ & =1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s}) \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6.9284em","vertical-align":"-3.2142em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k")])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.7142em"}},[s("span",{style:{top:"-5.7142em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"0.40"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-3.0512em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.2116em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8984em"}},[s("span",{style:{top:"-3.1473em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"L"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"0.16"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathrm"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0384em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-0.8728em"}},[s("span",{class:"pstrut",style:{height:"3.427em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.2142em"}},[s("span")])])])])])])])])])])],-1),g=s("p",null,[a("So, the value of the rate constant for the reaction is "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1.0"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"l")]),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"s"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"1.0 \\mathrm{~L} /(\\mathrm{mol} \\cdot \\mathrm{s})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.0"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mol")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"s"),s("span",{class:"mclose"},")")])])])],-1),y=[e,r,i,c,p,o,h,d,u,g];function v(x,w,k,b,f,L){return m(),t("div",null,y)}const z=l(n,[["render",v]]);export{_ as __pageData,z as default}; diff --git a/assets/academic_chemistry_problems_03-02-3.md.90fcd861.js b/assets/academic_chemistry_problems_03-02-3.md.2d09c751.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-3.md.90fcd861.js rename to assets/academic_chemistry_problems_03-02-3.md.2d09c751.js index f577ac23..48252f74 100644 --- a/assets/academic_chemistry_problems_03-02-3.md.90fcd861.js +++ b/assets/academic_chemistry_problems_03-02-3.md.2d09c751.js @@ -1 +1 @@ -import{_ as t,c as e,o as l,z as a,a as s,R as n}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1684858945000}'),m={name:"academic/chemistry/problems/03-02-3.md"},r=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),c=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),o=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),h=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),p=n('

Solution ​

',2),d=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),g=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1),u=[r,c,i,o,h,p,d,g];function x(w,y,k,b,f,M){return l(),e("div",null,u)}const X=t(m,[["render",x]]);export{_ as __pageData,X as default}; +import{_ as t,c as e,o as l,z as a,a as s,R as n}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1684859551000}'),m={name:"academic/chemistry/problems/03-02-3.md"},r=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),c=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),o=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),h=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),p=n('

Solution ​

',2),d=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),g=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1),u=[r,c,i,o,h,p,d,g];function x(w,y,k,b,f,M){return l(),e("div",null,u)}const X=t(m,[["render",x]]);export{_ as __pageData,X as default}; diff --git a/assets/academic_chemistry_problems_03-02-3.md.90fcd861.lean.js b/assets/academic_chemistry_problems_03-02-3.md.2d09c751.lean.js similarity index 99% rename from assets/academic_chemistry_problems_03-02-3.md.90fcd861.lean.js rename to assets/academic_chemistry_problems_03-02-3.md.2d09c751.lean.js index f352cf9e..0e0bcc06 100644 --- a/assets/academic_chemistry_problems_03-02-3.md.90fcd861.lean.js +++ b/assets/academic_chemistry_problems_03-02-3.md.2d09c751.lean.js @@ -1 +1 @@ -import{_ as t,c as e,o as l,z as a,a as s,R as n}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1684858945000}'),m={name:"academic/chemistry/problems/03-02-3.md"},r=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),c=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),o=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),h=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),p=n("",2),d=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),g=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1),u=[r,c,i,o,h,p,d,g];function x(w,y,k,b,f,M){return l(),e("div",null,u)}const X=t(m,[["render",x]]);export{_ as __pageData,X as default}; +import{_ as t,c as e,o as l,z as a,a as s,R as n}from"./chunks/framework.a1bac067.js";const _=JSON.parse('{"title":"Problem 03-02-3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/chemistry/problems/03-02-3.md","filePath":"academic/chemistry/problems/03-02-3.md","lastUpdated":1684859551000}'),m={name:"academic/chemistry/problems/03-02-3.md"},r=a("h1",{id:"problem-03-02-3",tabindex:"-1"},[s("Problem 03-02-3 "),a("a",{class:"header-anchor",href:"#problem-03-02-3","aria-label":'Permalink to "Problem 03-02-3"'},"​")],-1),c=a("h2",{id:"question",tabindex:"-1"},[s("Question "),a("a",{class:"header-anchor",href:"#question","aria-label":'Permalink to "Question"'},"​")],-1),i=a("p",{class:"katex-block"},[a("span",{class:"katex-display"},[a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("semantics",null,[a("mrow",null,[a("mn",null,"2"),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"→"),a("mi",{mathvariant:"normal"},"Q"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mn",null,"2"),a("mi",{mathvariant:"normal"},"R"),a("mo",{stretchy:"false"},"("),a("mi",null,"g"),a("mo",{stretchy:"false"},")")]),a("annotation",{encoding:"application/x-tex"},"2 \\mathrm{X}(g)+2 \\mathrm{Y}(g) \\rightarrow \\mathrm{Q}(g)+2 \\mathrm{R}(g) ")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),a("span",{class:"mrel"},"→"),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathrm"},"Q"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),a("span",{class:"mbin"},"+"),a("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mord"},"2"),a("span",{class:"mord mathrm"},"R"),a("span",{class:"mopen"},"("),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),a("span",{class:"mclose"},")")])])])])],-1),o=a("p",null,[s("The reaction represented above is found to be second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(".")],-1),h=a("p",null,[s("What happens to the rate of the reaction when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled?")],-1),p=n("",2),d=a("p",null,[s("To solve this problem, let's first write out the rate law for the reaction. According to the text, the reaction is second order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"X")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm"},"X")])])]),s(" and first order with respect to "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"Y")]),a("annotation",{encoding:"application/x-tex"},"\\mathrm{Y}")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.6833em"}}),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y")])])]),s(", so the rate law is rate "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",null,"="),a("mi",null,"k"),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("msup",null,[a("mo",{stretchy:"false"},"]"),a("mn",null,"2")]),a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"=k[\\mathrm{X}]^2[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.3669em"}}),a("span",{class:"mrel"},"="),a("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),a("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},[a("span",{class:"mclose"},"]"),a("span",{class:"msupsub"},[a("span",{class:"vlist-t"},[a("span",{class:"vlist-r"},[a("span",{class:"vlist",style:{height:"0.8141em"}},[a("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[a("span",{class:"pstrut",style:{height:"2.7em"}}),a("span",{class:"sizing reset-size6 size3 mtight"},[a("span",{class:"mord mtight"},"2")])])])])])])]),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])])],-1),g=a("p",null,[s("Now, let's think about how the rate changes when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled. Since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the second power in the rate law, halving "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" decreases the reaction rate by a factor of 4 . Similarly, since "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is raised to the first power, doubling "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" increases the reaction rate by a factor of 2 . Combined, these changes result in the reaction rate decreasing by a factor of 2 overall. So, when "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"X"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{X}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm"},"X"),a("span",{class:"mclose"},"]")])])]),s(" is halved and "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mo",{stretchy:"false"},"["),a("mi",{mathvariant:"normal"},"Y"),a("mo",{stretchy:"false"},"]")]),a("annotation",{encoding:"application/x-tex"},"[\\mathrm{Y}]")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),a("span",{class:"mopen"},"["),a("span",{class:"mord mathrm",style:{"margin-right":"0.025em"}},"Y"),a("span",{class:"mclose"},"]")])])]),s(" is doubled, the rate of the reaction decreases by a factor of 2 .")],-1),u=[r,c,i,o,h,p,d,g];function x(w,y,k,b,f,M){return l(),e("div",null,u)}const X=t(m,[["render",x]]);export{_ as __pageData,X as default}; diff --git a/assets/academic_literature_index.md.e8ed8386.js b/assets/academic_literature_index.md.b9024d79.js similarity index 90% rename from assets/academic_literature_index.md.e8ed8386.js rename to assets/academic_literature_index.md.b9024d79.js index eccfc626..9fcb8172 100644 --- a/assets/academic_literature_index.md.e8ed8386.js +++ b/assets/academic_literature_index.md.b9024d79.js @@ -1 +1 @@ -import{_ as t,c as a,o as r,z as e,a as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Welcome to Literature","description":"","frontmatter":{},"headers":[],"relativePath":"academic/literature/index.md","filePath":"academic/literature/index.md","lastUpdated":1684858945000}'),c={name:"academic/literature/index.md"},i=e("h1",{id:"welcome-to-literature",tabindex:"-1"},[o("Welcome to Literature "),e("a",{class:"header-anchor",href:"#welcome-to-literature","aria-label":'Permalink to "Welcome to Literature"'},"​")],-1),d=[i];function s(n,l,m,_,u,p){return r(),a("div",null,d)}const x=t(c,[["render",s]]);export{f as __pageData,x as default}; +import{_ as t,c as a,o as r,z as e,a as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Welcome to Literature","description":"","frontmatter":{},"headers":[],"relativePath":"academic/literature/index.md","filePath":"academic/literature/index.md","lastUpdated":1684859551000}'),c={name:"academic/literature/index.md"},i=e("h1",{id:"welcome-to-literature",tabindex:"-1"},[o("Welcome to Literature "),e("a",{class:"header-anchor",href:"#welcome-to-literature","aria-label":'Permalink to "Welcome to Literature"'},"​")],-1),d=[i];function s(n,l,m,_,u,p){return r(),a("div",null,d)}const x=t(c,[["render",s]]);export{f as __pageData,x as default}; diff --git a/assets/academic_literature_index.md.e8ed8386.lean.js b/assets/academic_literature_index.md.b9024d79.lean.js similarity index 90% rename from assets/academic_literature_index.md.e8ed8386.lean.js rename to assets/academic_literature_index.md.b9024d79.lean.js index eccfc626..9fcb8172 100644 --- a/assets/academic_literature_index.md.e8ed8386.lean.js +++ b/assets/academic_literature_index.md.b9024d79.lean.js @@ -1 +1 @@ -import{_ as t,c as a,o as r,z as e,a as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Welcome to Literature","description":"","frontmatter":{},"headers":[],"relativePath":"academic/literature/index.md","filePath":"academic/literature/index.md","lastUpdated":1684858945000}'),c={name:"academic/literature/index.md"},i=e("h1",{id:"welcome-to-literature",tabindex:"-1"},[o("Welcome to Literature "),e("a",{class:"header-anchor",href:"#welcome-to-literature","aria-label":'Permalink to "Welcome to Literature"'},"​")],-1),d=[i];function s(n,l,m,_,u,p){return r(),a("div",null,d)}const x=t(c,[["render",s]]);export{f as __pageData,x as default}; +import{_ as t,c as a,o as r,z as e,a as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Welcome to Literature","description":"","frontmatter":{},"headers":[],"relativePath":"academic/literature/index.md","filePath":"academic/literature/index.md","lastUpdated":1684859551000}'),c={name:"academic/literature/index.md"},i=e("h1",{id:"welcome-to-literature",tabindex:"-1"},[o("Welcome to Literature "),e("a",{class:"header-anchor",href:"#welcome-to-literature","aria-label":'Permalink to "Welcome to Literature"'},"​")],-1),d=[i];function s(n,l,m,_,u,p){return r(),a("div",null,d)}const x=t(c,[["render",s]]);export{f as __pageData,x as default}; diff --git a/assets/academic_literature_writing_methods-of-development.md.cb0d57b6.js b/assets/academic_literature_writing_methods-of-development.md.b79948e6.js similarity index 99% rename from assets/academic_literature_writing_methods-of-development.md.cb0d57b6.js rename to assets/academic_literature_writing_methods-of-development.md.b79948e6.js index d25e6cd5..e66a0095 100644 --- a/assets/academic_literature_writing_methods-of-development.md.cb0d57b6.js +++ b/assets/academic_literature_writing_methods-of-development.md.b79948e6.js @@ -1 +1 @@ -import{_ as e,c as t,o,R as a}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review","frontmatter":{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review"},"headers":[],"relativePath":"academic/literature/writing/methods-of-development.md","filePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1684858945000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a('

Patterns of Organization and Methods of Development ​

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

',41),r=[n];function s(l,h,d,c,p,u){return o(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default}; +import{_ as e,c as t,o,R as a}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review","frontmatter":{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review"},"headers":[],"relativePath":"academic/literature/writing/methods-of-development.md","filePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1684859551000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a('

Patterns of Organization and Methods of Development ​

Patterns of organization can help your readers follow the ideas within your essay and your paragraphs, but they can also work as methods of development to help you recognize and further develop ideas and relationships in your writing. Here are some strategies that can help you with both organization and development in your essays.

Major Patterns of Organization ​

A fruit pie.

Read the following sentences:

How did it feel to read the above list? A bit confusing, I would guess. That’s because the steps for making a pie were not well organized, and the steps don’t include enough detail for us to know exactly what we should do. (Like what are the dry and liquid ingredients?) We all know that starting instructions from the beginning and giving each detailed step in the order it should happen is vital to having a good outcome, in this case a yummy pie! But it’s not always so simple to know how to organize or develop ideas, and sometimes there’s more than one way, which complicates things even further.

First, let’s take a look at a couple of ways to think about organization.

General to Specific or Specific to General ​

It might be useful to think about organizing your topic like a triangle:

Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General.

The first triangle represents starting with the most general, big picture information first, moving then to more detailed and often more personal information later in the paper. The second triangle represents an organizational structure that starts with the specific, small scale information first and then moves to the more global, big picture stuff.

For example, if your topic is traffic in Vancouver, British Columbia, an essay that uses the general-to-specific organizational structure might begin this way:

Many people consider Vancouver, British Columbia, to be a relaxed place to live. They would be shocked to know how bad the traffic is traveling major arteries into the city and even driving around the city itself.

An essay that uses the specific-to-general structure might start like this:

Transit is crowded, parking is expensive, and vehicles stop and go through the main streets of the city of Vancouver, British Columbia, and that is just once travelers brave the crowded arteries to enter the city; Vancouver’s traffic problem does not lend itself to the relaxed atmosphere many believe the city to have.

What’s the difference between these two introductions? And how might they appeal to the intended audience for this essay in different ways? The first introduction is looking at the big picture of the problem and mentions pollution’s impact on all citizens in Portland, while the second introduction focuses on one specific family. The first helps readers see how vast the problem really is, and the second helps connect readers to a real family, making an emotional appeal from the very beginning. Neither introduction is necessarily better. You’ll choose one over the other based on the kind of tone you’d like to create and how you’d like to affect your audience. It’s completely up to you to make this decision.

Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? ​

The triangle is kind of a general guide, meaning you’re allowed to move around within it all you want. For example, it’s possible that each of your paragraphs will be its own triangle, starting with the general or specific and moving out or in. However, if you begin very broadly, it might be effective to end your essay in a more specific, personal way. And if you begin with a personal story, consider ending your essay by touching on the global impact and importance of your topic.

Are There Other Ways to Think about Organizing My Ideas? ​

Yes! Rather than thinking about which of your ideas are most specific or personal or which are more broad or universal, you might consider one of the following ways of organizing your ideas:

The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.

Choose one of the following topics, and practice writing a few opening sentences like we did above, once using the general-to-specific format and once using the specific-to-general. Which do you like better? What audience would be attracted to which one? Share with peers to see how others tackled this challenge. How would you rewrite their sentences? Why? Discuss your changes and listen to how your peers have revised your sentences. Taking in other people’s ideas will help you see new ways to approach your own writing and thinking.
Topics:

Methods of Development ​

The methods of development covered here are best used as ways to look at what’s already happening in your draft and to consider how you might emphasize or expand on any existing patterns. You might already be familiar with some of these patterns because teachers will sometimes assign them as the purpose for writing an essay. For example, you might have been asked to write a cause-and-effect essay or a comparison-and-contrast essay.

It’s important to emphasize here that patterns of organization or methods of developing content usually happen naturally as a consequence of the way the writer engages with and organizes information while writing. That is to say, most writers don’t sit down and say, “I think I’ll write a cause-and-effect essay today.” Instead, a writer might be more likely to be interested in a topic, say, the state of drinking water in the local community, and as the writer begins to explore the topic, certain cause-and-effect relationships between environmental pollutants and the community water supply may begin to emerge.

So if these patterns just occur naturally in writing, what’s the use in knowing about them? Well, sometimes you might be revising a draft and notice that some of your paragraphs are a bit underdeveloped. Maybe they lack a clear topic, or maybe they lack support. In either case, you can look to these common methods of development to find ways to sharpen those vague topics or to add support where needed. Do you have a clear cause statement somewhere but you haven’t explored the effects? Are you lacking detail somewhere where a narrative story or historical chronology can help build reader interest and add support? Are you struggling to define an idea that might benefit from some comparison or contrast? Read on to consider some of the ways that these strategies can help you in revision.

Cause and Effect (or Effect and Cause) ​

Do you see a potential cause-and-effect relationship developing in your draft? The cause-and-effect pattern may be used to identify one or more causes followed by one or more effects or results. Or you may reverse this sequence and describe effects first and then the cause or causes. For example, the causes of water pollution might be followed by its effects on both humans and animals. You may use obvious transitions to clarify cause and effect, such as “What are the results? Here are some of them…” or you might simply use the words cause, effect, and result, to cue the reader about your about the relationships that you’re establishing.

Problem-Solution ​

At some point does your essay explore a problem or suggest a solution? The problem-solution pattern is commonly used in identifying something that’s wrong and in contemplating what might be done to remedy the situation. There are probably more ways to organize a problem-solution approach, but but here are three possibilities:

When the solution is stated at the end of the paper, the pattern is sometimes called the delayed proposal. For a hostile audience, it may be effective to describe the problem, show why other solutions do not work, and finally suggest the favored solution. You can emphasize the words problem and solution to signal these sections of your paper for your reader.

Chronology or Narrative ​

Do you need to develop support for a topic where telling a story can illustrate some important concept for your readers? Material arranged chronologically is explained as it occurs in time. A chronological or narrative method of development might help you find a way to add both interest and content to your essay. Material arranged chronologically is explained as it occurs in time. This pattern may be used to establish what has happened. Chronology or narrative can be a great way to introduce your essay by providing a background or history behind your topic. Or you may want to tell a story to develop one or more points in the body of your essay. You can use transitional words like then, next, and finally to make the parts of the chronology clear.

Comparison and Contrast ​

Are you trying to define something? Do you need your readers to understand what something is and what it is not? The comparison-and-contrast method of development is particularly useful in extending a definition, or anywhere you need to show how a subject is like or unlike another subject. For example, the statement is often made that drug abuse is a medical problem instead of a criminal justice issue. An author might attempt to prove this point by comparing drug addiction to AIDS, cancer, or heart disease to redefine the term “addiction” as a medical problem. A statement in opposition to this idea could just as easily establish contrast by explaining all the ways that addiction is different from what we traditionally understand as an illness. In seeking to establish comparison or contrast in your writing, some words or terms that might be useful are by contrast, in comparison, while, some, and others.

Summary

These four methods of development—cause and effect, problem-solution, chronology or narrative, and comparison and contrast—are just a few ways to organize and develop ideas and content in your essays. It’s important to note that they should not be a starting point for writers who want to write something authentic—something that they care deeply about. Instead, they can be a great way to help you look for what’s already happening with your topic or in a draft, to help you to write more, or to help you reorganize some parts of an essay that seem to lack connection or feel disjointed. Look for organizational patterns when you’re reading work by professional writers. Notice where they combine strategies (e.g., a problem-solution pattern that uses cause-and-effect organization, or a comparison-contrast pattern that uses narrative or chronology to develop similarities or differences). Pay attention to how different writers emphasize and develop their main ideas, and use what you find to inspire you in your own writing. Better yet, work on developing completely new patterns of your own.

Reference

',41),r=[n];function s(l,h,d,c,p,u){return o(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default}; diff --git a/assets/academic_literature_writing_methods-of-development.md.cb0d57b6.lean.js b/assets/academic_literature_writing_methods-of-development.md.b79948e6.lean.js similarity index 91% rename from assets/academic_literature_writing_methods-of-development.md.cb0d57b6.lean.js rename to assets/academic_literature_writing_methods-of-development.md.b79948e6.lean.js index 07b6e29c..5f0cfe93 100644 --- a/assets/academic_literature_writing_methods-of-development.md.cb0d57b6.lean.js +++ b/assets/academic_literature_writing_methods-of-development.md.b79948e6.lean.js @@ -1 +1 @@ -import{_ as e,c as t,o,R as a}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review","frontmatter":{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review"},"headers":[],"relativePath":"academic/literature/writing/methods-of-development.md","filePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1684858945000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a("",41),r=[n];function s(l,h,d,c,p,u){return o(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default}; +import{_ as e,c as t,o,R as a}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review","frontmatter":{"title":"Patterns of Organization and Methods of Development","description":"Literature methods of development article clip for review"},"headers":[],"relativePath":"academic/literature/writing/methods-of-development.md","filePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1684859551000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a("",41),r=[n];function s(l,h,d,c,p,u){return o(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default}; diff --git a/assets/academic_physics_index.md.669c3eb1.js b/assets/academic_physics_index.md.837a8425.js similarity index 90% rename from assets/academic_physics_index.md.669c3eb1.js rename to assets/academic_physics_index.md.837a8425.js index 072e5573..17aeaa53 100644 --- a/assets/academic_physics_index.md.669c3eb1.js +++ b/assets/academic_physics_index.md.837a8425.js @@ -1 +1 @@ -import{_ as a,c as s,o as t,z as e,a as c}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Physics","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/index.md","filePath":"academic/physics/index.md","lastUpdated":1684858945000}'),o={name:"academic/physics/index.md"},i=e("h1",{id:"welcome-to-physics",tabindex:"-1"},[c("Welcome to Physics "),e("a",{class:"header-anchor",href:"#welcome-to-physics","aria-label":'Permalink to "Welcome to Physics"'},"​")],-1),d=[i];function n(r,l,h,m,p,_){return t(),s("div",null,d)}const y=a(o,[["render",n]]);export{x as __pageData,y as default}; +import{_ as a,c as s,o as t,z as e,a as c}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Physics","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/index.md","filePath":"academic/physics/index.md","lastUpdated":1684859551000}'),o={name:"academic/physics/index.md"},i=e("h1",{id:"welcome-to-physics",tabindex:"-1"},[c("Welcome to Physics "),e("a",{class:"header-anchor",href:"#welcome-to-physics","aria-label":'Permalink to "Welcome to Physics"'},"​")],-1),d=[i];function n(r,l,h,m,p,_){return t(),s("div",null,d)}const y=a(o,[["render",n]]);export{x as __pageData,y as default}; diff --git a/assets/academic_physics_index.md.669c3eb1.lean.js b/assets/academic_physics_index.md.837a8425.lean.js similarity index 90% rename from assets/academic_physics_index.md.669c3eb1.lean.js rename to assets/academic_physics_index.md.837a8425.lean.js index 072e5573..17aeaa53 100644 --- a/assets/academic_physics_index.md.669c3eb1.lean.js +++ b/assets/academic_physics_index.md.837a8425.lean.js @@ -1 +1 @@ -import{_ as a,c as s,o as t,z as e,a as c}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Physics","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/index.md","filePath":"academic/physics/index.md","lastUpdated":1684858945000}'),o={name:"academic/physics/index.md"},i=e("h1",{id:"welcome-to-physics",tabindex:"-1"},[c("Welcome to Physics "),e("a",{class:"header-anchor",href:"#welcome-to-physics","aria-label":'Permalink to "Welcome to Physics"'},"​")],-1),d=[i];function n(r,l,h,m,p,_){return t(),s("div",null,d)}const y=a(o,[["render",n]]);export{x as __pageData,y as default}; +import{_ as a,c as s,o as t,z as e,a as c}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Welcome to Physics","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/index.md","filePath":"academic/physics/index.md","lastUpdated":1684859551000}'),o={name:"academic/physics/index.md"},i=e("h1",{id:"welcome-to-physics",tabindex:"-1"},[c("Welcome to Physics "),e("a",{class:"header-anchor",href:"#welcome-to-physics","aria-label":'Permalink to "Welcome to Physics"'},"​")],-1),d=[i];function n(r,l,h,m,p,_){return t(),s("div",null,d)}const y=a(o,[["render",n]]);export{x as __pageData,y as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.js b/assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.js rename to assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.js index ebe21570..144b3a90 100644 --- a/assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.js +++ b/assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.js @@ -1,4 +1,4 @@ -import{_ as m,c as n,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const K=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/1.md","filePath":"academic/physics/ipho-formulas-jpn/1.md","lastUpdated":1684858945000}'),c={name:"academic/physics/ipho-formulas-jpn/1.md"},r=l('

Formulas for IPhO 日本語版: Section 1 ​

1: 数学 ​

1.1: Taylor 展開 ​

',3),h=s("ol",null,[s("li",null,[s("p",null,"Taylor 展開（アバウトに切り捨てる:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mo",null,"∑"),s("msup",null,[s("mi",null,"F"),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mo",{stretchy:"false"},")")])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mi",null,"n")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"F(x)=F\\left(x_{0}\\right)+\\sum F^{(n)}\\left(x_{0}\\right)\\left(x-x_{0}\\right)^{n} / n ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mclose mtight"},")")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8043em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"n")])])])])]),s("p",null,"線形近似（特別な場合）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"F"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"F(x) \\approx F\\left(x_{0}\\right)+F^{\\prime}\\left(x_{0}\\right)\\left(x-x_{0}\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0519em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8019em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])]),s("p",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"x"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"|x| \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" のときの例 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":")]),s("annotation",{encoding:"application/x-tex"},":")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"e"),s("mi",null,"x")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\sin x \\approx x, \\cos x \\approx 1-x^{2} / 2, e^{x} \\approx 1+x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x")])])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ln"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"n")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"n"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\ln (1+x) \\approx x,(1+x)^{n} \\approx 1+n x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"ln"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal"},"x")])])])])])])],-1),o=s("h3",{id:"_1-2-摂動法",tabindex:"-1"},[a("1.2: 摂動法 "),s("a",{class:"header-anchor",href:"#_1-2-摂動法","aria-label":'Permalink to "1.2: 摂動法"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("摂動法：摂動のない（直接解ける）問題の解を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" 番目の近似値として求め，前の似値に基づく次の近似値の補正を繰り返して解を求める．")])],-1),u=s("h3",{id:"_1-3-定数係数線形微分方程式",tabindex:"-1"},[a("1.3: 定数係数線形微分方程式 "),s("a",{class:"header-anchor",href:"#_1-3-定数係数線形微分方程式","aria-label":'Permalink to "1.3: 定数係数線形微分方程式"'},"​")],-1),y=s("ol",{start:"3"},[s("li",null,[s("p",null,[a("定数係数線形微分方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"y"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])]),s("mo",null,"+"),s("mi",null,"b"),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"+"),s("mi",null,"c"),s("mi",null,"y"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a y^{\\prime \\prime}+b y^{\\prime}+c y=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"cy"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の解:")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"A"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"1")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mi",null,"B"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"y=A \\exp \\left(\\lambda_1 x\\right)+B \\exp \\left(\\lambda_2 x\\right) \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])]),s("p",null,[a("ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" は特性方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"λ"),s("mo",null,"+"),s("mi",null,"c"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a \\lambda^2+b \\lambda+c=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"bλ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の異な る 2 解. もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a, b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" が実数で特性方程式の解が複素数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])]),s("mo",null,"="),s("mi",null,"γ"),s("mo",null,"±"),s("mi",null,"i"),s("mi",null,"ω")]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}=\\gamma \\pm i \\omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω")])])]),a(" ならば,")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"C"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"γ"),s("mi",null,"x")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"ω"),s("mi",null,"x"),s("mo",null,"+"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y=C e^{\\gamma x} \\sin \\left(\\omega x+\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])])])],-1),d=s("h3",{id:"_1-4-複素数",tabindex:"-1"},[a("1.4: 複素数 "),s("a",{class:"header-anchor",href:"#_1-4-複素数","aria-label":'Permalink to "1.4: 複素数"'},"​")],-1),v=s("ol",{start:"4"},[s("li",null,[s("p",null,"複素数"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"z"),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",{separator:"true"},","),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"−"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("msup",null,[s("mi",{mathvariant:"normal"},"∣"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"z"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mfrac",null,[s("mi",null,"b"),s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"Re"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"+"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Im"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"−"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"i")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",{fence:"true"},"∣")]),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",{separator:"true"},","),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",null,"+"),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"+"),s("mi",null,"i"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"+"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"−"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"i")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} z=a+b i=|z| e^{i \\varphi}, \\bar{z}=a-b i=|z| e^{-i \\varphi} \\\\ |z|^2=z \\bar{z}=a^2+b^2, \\varphi=\\arg z=\\arcsin \\frac{b}{|z|} \\\\ \\operatorname{Re} z=(z+\\bar{z}) / 2, \\operatorname{Im} z=(z-\\bar{z}) / 2 i \\\\ \\left|z_1 z_2\\right|=\\left|z_1\\right|\\left|z_2\\right|, \\arg z_1 z_2=\\arg z_1+\\arg z_2 \\\\ e^{i \\varphi}=\\cos \\varphi+i \\sin \\varphi \\\\ \\cos \\varphi=\\frac{e^{i \\varphi}+e^{-i \\varphi}}{2}, \\sin \\varphi=\\frac{e^{i \\varphi}-e^{-i \\varphi}}{2 i} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.1644em","vertical-align":"-5.3322em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8322em"}},[s("span",{style:{top:"-8.4592em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])]),s("span",{style:{top:"-6.4278em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-4.3518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Re")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Im")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-2.8518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.3171em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"0.8446em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.3322em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_1-5-ベクトルの内積と外積",tabindex:"-1"},[a("1.5: ベクトルの内積と外積 "),s("a",{class:"header-anchor",href:"#_1-5-ベクトルの内積と外積","aria-label":'Permalink to "1.5: ベクトルの内積と外積"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[s("p",null,[a("ベクトルの内積と外積は分配法則が成立する : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"a"),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a(b+c)=a b+a c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mo",null,"−"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⊥"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",null,"+"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"−"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"c")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{a} \\cdot \\boldsymbol{b}=\\boldsymbol{b} \\cdot \\boldsymbol{a}=a_x b_x+a_y b_y+\\cdots=a b \\cos \\varphi \\\\ |\\boldsymbol{a} \\times \\boldsymbol{b}|=a b \\sin \\varphi, \\boldsymbol{a} \\times \\boldsymbol{b}=-\\boldsymbol{b} \\times \\boldsymbol{a} \\perp \\boldsymbol{a}, \\boldsymbol{b} \\\\ \\boldsymbol{a} \\times \\boldsymbol{b}=\\left(a_y b_z-a_z b_y\\right) \\boldsymbol{e}_x+\\left(a_z b_x-a_x b_z\\right) \\boldsymbol{e}_y+\\cdots \\\\ \\boldsymbol{a} \\times[\\boldsymbol{b} \\times \\boldsymbol{c}]=(\\boldsymbol{a} \\cdot \\boldsymbol{c}) \\boldsymbol{b}-(\\boldsymbol{a} \\cdot \\boldsymbol{b}) \\boldsymbol{c} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6em","vertical-align":"-2.75em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.25em"}},[s("span",{style:{top:"-5.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"-3.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯")])]),s("span",{style:{top:"-0.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.75em"}},[s("span")])])])])])])])])])])]),s("p",null,"スカラー三重積（3 つのベクトルで張られる平行四面 体の体積）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"≡"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},"]"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}) \\equiv \\boldsymbol{a} \\cdot[\\boldsymbol{b} \\times \\boldsymbol{c}]=[\\boldsymbol{a} \\times \\boldsymbol{b}] \\cdot \\boldsymbol{c}=(\\boldsymbol{b}, \\boldsymbol{c}, \\boldsymbol{a}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4445em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mclose"},")")])])])])])])],-1),z=s("h3",{id:"_1-6-余弦定理と正弦定理",tabindex:"-1"},[a("1.6: 余弦定理と正弦定理 "),s("a",{class:"header-anchor",href:"#_1-6-余弦定理と正弦定理","aria-label":'Permalink to "1.6: 余弦定理と正弦定理"'},"​")],-1),f=s("ol",{start:"6"},[s("li",null,[a("余弦定理と正弦定理："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"C")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"a"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"A"),s("mo",null,"="),s("mi",null,"b"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"B"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & c^2=a^2+b^2-2 a b \\cos C \\\\ & a / \\sin A=b / \\sin B=2 R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.7621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.2621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_1-7-三角法",tabindex:"-1"},[a("1.7: 三角法 "),s("a",{class:"header-anchor",href:"#_1-7-三角法","aria-label":'Permalink to "1.7: 三角法"'},"​")],-1),_=s("ol",{start:"7"},[s("li",null,[s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"±"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"∓"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"∓"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mn",null,"2"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\sin (\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\\\ & \\cos (\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta \\\\ & \\tan (\\alpha \\pm \\beta)=(\\tan \\alpha \\pm \\tan \\beta) /(1 \\mp \\tan \\alpha \\tan \\beta) \\\\ & \\cos ^2 \\alpha=\\frac{1+\\cos 2 \\alpha}{2}, \\sin ^2 \\alpha=\\frac{1-\\cos 2 \\alpha}{2} \\\\ & \\cos \\alpha \\cos \\beta=\\frac{\\cos (\\alpha+\\beta)+\\cos (\\alpha-\\beta)}{2}, \\ldots \\\\ & \\cos \\alpha+\\cos \\beta=2\\left(\\cos \\frac{\\alpha+\\beta}{2}+\\cos \\frac{\\alpha-\\beta}{2}\\right), \\ldots \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.9205em","vertical-align":"-5.7102em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])])])])])])])])])])],-1),k=l('

1.8: 円周角 ​

  1. 円周角は中心角の半分. よって，直角三角形の斜辺は その外接円の直径. もし四角形の対角の和が 180 度な らば，それは円に内接する.

1.9: 三角形の面樍 ​

',3),M=s("ol",{start:"9"},[s("li",null,[a("三角形の面樍 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"a"),s("msub",null,[s("mi",null,"h"),s("mi",null,"a")]),s("mo",null,"="),s("mi",null,"p"),s("mi",null,"r"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"p"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"a"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"c"),s("mo",{stretchy:"false"},")")])]),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"4"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"=\\frac{1}{2} a h_a=p r=\\sqrt{p(p-a)(p-b)(p-c)}=a b c / 4 R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"h"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as m,c as n,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const K=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/1.md","filePath":"academic/physics/ipho-formulas-jpn/1.md","lastUpdated":1684859551000}'),c={name:"academic/physics/ipho-formulas-jpn/1.md"},r=l('

Formulas for IPhO 日本語版: Section 1 ​

1: 数学 ​

1.1: Taylor 展開 ​

',3),h=s("ol",null,[s("li",null,[s("p",null,"Taylor 展開（アバウトに切り捨てる:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mo",null,"∑"),s("msup",null,[s("mi",null,"F"),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mo",{stretchy:"false"},")")])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mi",null,"n")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"F(x)=F\\left(x_{0}\\right)+\\sum F^{(n)}\\left(x_{0}\\right)\\left(x-x_{0}\\right)^{n} / n ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mclose mtight"},")")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8043em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"n")])])])])]),s("p",null,"線形近似（特別な場合）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"F"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"F(x) \\approx F\\left(x_{0}\\right)+F^{\\prime}\\left(x_{0}\\right)\\left(x-x_{0}\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0519em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8019em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])]),s("p",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"x"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"|x| \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" のときの例 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":")]),s("annotation",{encoding:"application/x-tex"},":")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"e"),s("mi",null,"x")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\sin x \\approx x, \\cos x \\approx 1-x^{2} / 2, e^{x} \\approx 1+x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x")])])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ln"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"n")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"n"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\ln (1+x) \\approx x,(1+x)^{n} \\approx 1+n x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"ln"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal"},"x")])])])])])])],-1),o=s("h3",{id:"_1-2-摂動法",tabindex:"-1"},[a("1.2: 摂動法 "),s("a",{class:"header-anchor",href:"#_1-2-摂動法","aria-label":'Permalink to "1.2: 摂動法"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("摂動法：摂動のない（直接解ける）問題の解を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" 番目の近似値として求め，前の似値に基づく次の近似値の補正を繰り返して解を求める．")])],-1),u=s("h3",{id:"_1-3-定数係数線形微分方程式",tabindex:"-1"},[a("1.3: 定数係数線形微分方程式 "),s("a",{class:"header-anchor",href:"#_1-3-定数係数線形微分方程式","aria-label":'Permalink to "1.3: 定数係数線形微分方程式"'},"​")],-1),y=s("ol",{start:"3"},[s("li",null,[s("p",null,[a("定数係数線形微分方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"y"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])]),s("mo",null,"+"),s("mi",null,"b"),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"+"),s("mi",null,"c"),s("mi",null,"y"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a y^{\\prime \\prime}+b y^{\\prime}+c y=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"cy"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の解:")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"A"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"1")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mi",null,"B"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"y=A \\exp \\left(\\lambda_1 x\\right)+B \\exp \\left(\\lambda_2 x\\right) \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])]),s("p",null,[a("ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" は特性方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"λ"),s("mo",null,"+"),s("mi",null,"c"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a \\lambda^2+b \\lambda+c=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"bλ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の異な る 2 解. もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a, b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" が実数で特性方程式の解が複素数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])]),s("mo",null,"="),s("mi",null,"γ"),s("mo",null,"±"),s("mi",null,"i"),s("mi",null,"ω")]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}=\\gamma \\pm i \\omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω")])])]),a(" ならば,")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"C"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"γ"),s("mi",null,"x")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"ω"),s("mi",null,"x"),s("mo",null,"+"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y=C e^{\\gamma x} \\sin \\left(\\omega x+\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])])])],-1),d=s("h3",{id:"_1-4-複素数",tabindex:"-1"},[a("1.4: 複素数 "),s("a",{class:"header-anchor",href:"#_1-4-複素数","aria-label":'Permalink to "1.4: 複素数"'},"​")],-1),v=s("ol",{start:"4"},[s("li",null,[s("p",null,"複素数"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"z"),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",{separator:"true"},","),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"−"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("msup",null,[s("mi",{mathvariant:"normal"},"∣"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"z"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mfrac",null,[s("mi",null,"b"),s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"Re"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"+"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Im"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"−"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"i")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",{fence:"true"},"∣")]),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",{separator:"true"},","),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",null,"+"),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"+"),s("mi",null,"i"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"+"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"−"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"i")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} z=a+b i=|z| e^{i \\varphi}, \\bar{z}=a-b i=|z| e^{-i \\varphi} \\\\ |z|^2=z \\bar{z}=a^2+b^2, \\varphi=\\arg z=\\arcsin \\frac{b}{|z|} \\\\ \\operatorname{Re} z=(z+\\bar{z}) / 2, \\operatorname{Im} z=(z-\\bar{z}) / 2 i \\\\ \\left|z_1 z_2\\right|=\\left|z_1\\right|\\left|z_2\\right|, \\arg z_1 z_2=\\arg z_1+\\arg z_2 \\\\ e^{i \\varphi}=\\cos \\varphi+i \\sin \\varphi \\\\ \\cos \\varphi=\\frac{e^{i \\varphi}+e^{-i \\varphi}}{2}, \\sin \\varphi=\\frac{e^{i \\varphi}-e^{-i \\varphi}}{2 i} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.1644em","vertical-align":"-5.3322em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8322em"}},[s("span",{style:{top:"-8.4592em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])]),s("span",{style:{top:"-6.4278em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-4.3518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Re")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Im")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-2.8518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.3171em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"0.8446em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.3322em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_1-5-ベクトルの内積と外積",tabindex:"-1"},[a("1.5: ベクトルの内積と外積 "),s("a",{class:"header-anchor",href:"#_1-5-ベクトルの内積と外積","aria-label":'Permalink to "1.5: ベクトルの内積と外積"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[s("p",null,[a("ベクトルの内積と外積は分配法則が成立する : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"a"),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a(b+c)=a b+a c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mo",null,"−"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⊥"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",null,"+"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"−"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"c")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{a} \\cdot \\boldsymbol{b}=\\boldsymbol{b} \\cdot \\boldsymbol{a}=a_x b_x+a_y b_y+\\cdots=a b \\cos \\varphi \\\\ |\\boldsymbol{a} \\times \\boldsymbol{b}|=a b \\sin \\varphi, \\boldsymbol{a} \\times \\boldsymbol{b}=-\\boldsymbol{b} \\times \\boldsymbol{a} \\perp \\boldsymbol{a}, \\boldsymbol{b} \\\\ \\boldsymbol{a} \\times \\boldsymbol{b}=\\left(a_y b_z-a_z b_y\\right) \\boldsymbol{e}_x+\\left(a_z b_x-a_x b_z\\right) \\boldsymbol{e}_y+\\cdots \\\\ \\boldsymbol{a} \\times[\\boldsymbol{b} \\times \\boldsymbol{c}]=(\\boldsymbol{a} \\cdot \\boldsymbol{c}) \\boldsymbol{b}-(\\boldsymbol{a} \\cdot \\boldsymbol{b}) \\boldsymbol{c} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6em","vertical-align":"-2.75em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.25em"}},[s("span",{style:{top:"-5.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"-3.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯")])]),s("span",{style:{top:"-0.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.75em"}},[s("span")])])])])])])])])])])]),s("p",null,"スカラー三重積（3 つのベクトルで張られる平行四面 体の体積）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"≡"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},"]"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}) \\equiv \\boldsymbol{a} \\cdot[\\boldsymbol{b} \\times \\boldsymbol{c}]=[\\boldsymbol{a} \\times \\boldsymbol{b}] \\cdot \\boldsymbol{c}=(\\boldsymbol{b}, \\boldsymbol{c}, \\boldsymbol{a}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4445em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mclose"},")")])])])])])])],-1),z=s("h3",{id:"_1-6-余弦定理と正弦定理",tabindex:"-1"},[a("1.6: 余弦定理と正弦定理 "),s("a",{class:"header-anchor",href:"#_1-6-余弦定理と正弦定理","aria-label":'Permalink to "1.6: 余弦定理と正弦定理"'},"​")],-1),f=s("ol",{start:"6"},[s("li",null,[a("余弦定理と正弦定理："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"C")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"a"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"A"),s("mo",null,"="),s("mi",null,"b"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"B"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & c^2=a^2+b^2-2 a b \\cos C \\\\ & a / \\sin A=b / \\sin B=2 R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.7621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.2621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_1-7-三角法",tabindex:"-1"},[a("1.7: 三角法 "),s("a",{class:"header-anchor",href:"#_1-7-三角法","aria-label":'Permalink to "1.7: 三角法"'},"​")],-1),_=s("ol",{start:"7"},[s("li",null,[s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"±"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"∓"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"∓"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mn",null,"2"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\sin (\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\\\ & \\cos (\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta \\\\ & \\tan (\\alpha \\pm \\beta)=(\\tan \\alpha \\pm \\tan \\beta) /(1 \\mp \\tan \\alpha \\tan \\beta) \\\\ & \\cos ^2 \\alpha=\\frac{1+\\cos 2 \\alpha}{2}, \\sin ^2 \\alpha=\\frac{1-\\cos 2 \\alpha}{2} \\\\ & \\cos \\alpha \\cos \\beta=\\frac{\\cos (\\alpha+\\beta)+\\cos (\\alpha-\\beta)}{2}, \\ldots \\\\ & \\cos \\alpha+\\cos \\beta=2\\left(\\cos \\frac{\\alpha+\\beta}{2}+\\cos \\frac{\\alpha-\\beta}{2}\\right), \\ldots \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.9205em","vertical-align":"-5.7102em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])])])])])])])])])])],-1),k=l('

1.8: 円周角 ​

  1. 円周角は中心角の半分. よって，直角三角形の斜辺は その外接円の直径. もし四角形の対角の和が 180 度な らば，それは円に内接する.

1.9: 三角形の面樍 ​

',3),M=s("ol",{start:"9"},[s("li",null,[a("三角形の面樍 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"a"),s("msub",null,[s("mi",null,"h"),s("mi",null,"a")]),s("mo",null,"="),s("mi",null,"p"),s("mi",null,"r"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"p"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"a"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"c"),s("mo",{stretchy:"false"},")")])]),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"4"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"=\\frac{1}{2} a h_a=p r=\\sqrt{p(p-a)(p-b)(p-c)}=a b c / 4 R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"h"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.lean.js b/assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.lean.js rename to assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.lean.js index 3e046742..5a17d841 100644 --- a/assets/academic_physics_ipho-formulas-jpn_1.md.ec1dbaf7.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_1.md.1be1e0d1.lean.js @@ -1,4 +1,4 @@ -import{_ as m,c as n,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const K=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/1.md","filePath":"academic/physics/ipho-formulas-jpn/1.md","lastUpdated":1684858945000}'),c={name:"academic/physics/ipho-formulas-jpn/1.md"},r=l("",3),h=s("ol",null,[s("li",null,[s("p",null,"Taylor 展開（アバウトに切り捨てる:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mo",null,"∑"),s("msup",null,[s("mi",null,"F"),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mo",{stretchy:"false"},")")])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mi",null,"n")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"F(x)=F\\left(x_{0}\\right)+\\sum F^{(n)}\\left(x_{0}\\right)\\left(x-x_{0}\\right)^{n} / n ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mclose mtight"},")")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8043em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"n")])])])])]),s("p",null,"線形近似（特別な場合）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"F"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"F(x) \\approx F\\left(x_{0}\\right)+F^{\\prime}\\left(x_{0}\\right)\\left(x-x_{0}\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0519em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8019em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])]),s("p",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"x"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"|x| \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" のときの例 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":")]),s("annotation",{encoding:"application/x-tex"},":")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"e"),s("mi",null,"x")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\sin x \\approx x, \\cos x \\approx 1-x^{2} / 2, e^{x} \\approx 1+x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x")])])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ln"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"n")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"n"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\ln (1+x) \\approx x,(1+x)^{n} \\approx 1+n x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"ln"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal"},"x")])])])])])])],-1),o=s("h3",{id:"_1-2-摂動法",tabindex:"-1"},[a("1.2: 摂動法 "),s("a",{class:"header-anchor",href:"#_1-2-摂動法","aria-label":'Permalink to "1.2: 摂動法"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("摂動法：摂動のない（直接解ける）問題の解を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" 番目の近似値として求め，前の似値に基づく次の近似値の補正を繰り返して解を求める．")])],-1),u=s("h3",{id:"_1-3-定数係数線形微分方程式",tabindex:"-1"},[a("1.3: 定数係数線形微分方程式 "),s("a",{class:"header-anchor",href:"#_1-3-定数係数線形微分方程式","aria-label":'Permalink to "1.3: 定数係数線形微分方程式"'},"​")],-1),y=s("ol",{start:"3"},[s("li",null,[s("p",null,[a("定数係数線形微分方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"y"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])]),s("mo",null,"+"),s("mi",null,"b"),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"+"),s("mi",null,"c"),s("mi",null,"y"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a y^{\\prime \\prime}+b y^{\\prime}+c y=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"cy"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の解:")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"A"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"1")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mi",null,"B"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"y=A \\exp \\left(\\lambda_1 x\\right)+B \\exp \\left(\\lambda_2 x\\right) \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])]),s("p",null,[a("ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" は特性方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"λ"),s("mo",null,"+"),s("mi",null,"c"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a \\lambda^2+b \\lambda+c=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"bλ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の異な る 2 解. もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a, b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" が実数で特性方程式の解が複素数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])]),s("mo",null,"="),s("mi",null,"γ"),s("mo",null,"±"),s("mi",null,"i"),s("mi",null,"ω")]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}=\\gamma \\pm i \\omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω")])])]),a(" ならば,")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"C"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"γ"),s("mi",null,"x")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"ω"),s("mi",null,"x"),s("mo",null,"+"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y=C e^{\\gamma x} \\sin \\left(\\omega x+\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])])])],-1),d=s("h3",{id:"_1-4-複素数",tabindex:"-1"},[a("1.4: 複素数 "),s("a",{class:"header-anchor",href:"#_1-4-複素数","aria-label":'Permalink to "1.4: 複素数"'},"​")],-1),v=s("ol",{start:"4"},[s("li",null,[s("p",null,"複素数"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"z"),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",{separator:"true"},","),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"−"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("msup",null,[s("mi",{mathvariant:"normal"},"∣"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"z"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mfrac",null,[s("mi",null,"b"),s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"Re"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"+"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Im"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"−"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"i")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",{fence:"true"},"∣")]),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",{separator:"true"},","),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",null,"+"),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"+"),s("mi",null,"i"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"+"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"−"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"i")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} z=a+b i=|z| e^{i \\varphi}, \\bar{z}=a-b i=|z| e^{-i \\varphi} \\\\ |z|^2=z \\bar{z}=a^2+b^2, \\varphi=\\arg z=\\arcsin \\frac{b}{|z|} \\\\ \\operatorname{Re} z=(z+\\bar{z}) / 2, \\operatorname{Im} z=(z-\\bar{z}) / 2 i \\\\ \\left|z_1 z_2\\right|=\\left|z_1\\right|\\left|z_2\\right|, \\arg z_1 z_2=\\arg z_1+\\arg z_2 \\\\ e^{i \\varphi}=\\cos \\varphi+i \\sin \\varphi \\\\ \\cos \\varphi=\\frac{e^{i \\varphi}+e^{-i \\varphi}}{2}, \\sin \\varphi=\\frac{e^{i \\varphi}-e^{-i \\varphi}}{2 i} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.1644em","vertical-align":"-5.3322em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8322em"}},[s("span",{style:{top:"-8.4592em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])]),s("span",{style:{top:"-6.4278em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-4.3518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Re")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Im")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-2.8518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.3171em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"0.8446em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.3322em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_1-5-ベクトルの内積と外積",tabindex:"-1"},[a("1.5: ベクトルの内積と外積 "),s("a",{class:"header-anchor",href:"#_1-5-ベクトルの内積と外積","aria-label":'Permalink to "1.5: ベクトルの内積と外積"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[s("p",null,[a("ベクトルの内積と外積は分配法則が成立する : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"a"),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a(b+c)=a b+a c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mo",null,"−"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⊥"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",null,"+"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"−"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"c")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{a} \\cdot \\boldsymbol{b}=\\boldsymbol{b} \\cdot \\boldsymbol{a}=a_x b_x+a_y b_y+\\cdots=a b \\cos \\varphi \\\\ |\\boldsymbol{a} \\times \\boldsymbol{b}|=a b \\sin \\varphi, \\boldsymbol{a} \\times \\boldsymbol{b}=-\\boldsymbol{b} \\times \\boldsymbol{a} \\perp \\boldsymbol{a}, \\boldsymbol{b} \\\\ \\boldsymbol{a} \\times \\boldsymbol{b}=\\left(a_y b_z-a_z b_y\\right) \\boldsymbol{e}_x+\\left(a_z b_x-a_x b_z\\right) \\boldsymbol{e}_y+\\cdots \\\\ \\boldsymbol{a} \\times[\\boldsymbol{b} \\times \\boldsymbol{c}]=(\\boldsymbol{a} \\cdot \\boldsymbol{c}) \\boldsymbol{b}-(\\boldsymbol{a} \\cdot \\boldsymbol{b}) \\boldsymbol{c} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6em","vertical-align":"-2.75em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.25em"}},[s("span",{style:{top:"-5.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"-3.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯")])]),s("span",{style:{top:"-0.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.75em"}},[s("span")])])])])])])])])])])]),s("p",null,"スカラー三重積（3 つのベクトルで張られる平行四面 体の体積）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"≡"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},"]"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}) \\equiv \\boldsymbol{a} \\cdot[\\boldsymbol{b} \\times \\boldsymbol{c}]=[\\boldsymbol{a} \\times \\boldsymbol{b}] \\cdot \\boldsymbol{c}=(\\boldsymbol{b}, \\boldsymbol{c}, \\boldsymbol{a}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4445em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mclose"},")")])])])])])])],-1),z=s("h3",{id:"_1-6-余弦定理と正弦定理",tabindex:"-1"},[a("1.6: 余弦定理と正弦定理 "),s("a",{class:"header-anchor",href:"#_1-6-余弦定理と正弦定理","aria-label":'Permalink to "1.6: 余弦定理と正弦定理"'},"​")],-1),f=s("ol",{start:"6"},[s("li",null,[a("余弦定理と正弦定理："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"C")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"a"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"A"),s("mo",null,"="),s("mi",null,"b"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"B"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & c^2=a^2+b^2-2 a b \\cos C \\\\ & a / \\sin A=b / \\sin B=2 R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.7621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.2621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_1-7-三角法",tabindex:"-1"},[a("1.7: 三角法 "),s("a",{class:"header-anchor",href:"#_1-7-三角法","aria-label":'Permalink to "1.7: 三角法"'},"​")],-1),_=s("ol",{start:"7"},[s("li",null,[s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"±"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"∓"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"∓"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mn",null,"2"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\sin (\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\\\ & \\cos (\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta \\\\ & \\tan (\\alpha \\pm \\beta)=(\\tan \\alpha \\pm \\tan \\beta) /(1 \\mp \\tan \\alpha \\tan \\beta) \\\\ & \\cos ^2 \\alpha=\\frac{1+\\cos 2 \\alpha}{2}, \\sin ^2 \\alpha=\\frac{1-\\cos 2 \\alpha}{2} \\\\ & \\cos \\alpha \\cos \\beta=\\frac{\\cos (\\alpha+\\beta)+\\cos (\\alpha-\\beta)}{2}, \\ldots \\\\ & \\cos \\alpha+\\cos \\beta=2\\left(\\cos \\frac{\\alpha+\\beta}{2}+\\cos \\frac{\\alpha-\\beta}{2}\\right), \\ldots \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.9205em","vertical-align":"-5.7102em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])])])])])])])])])])],-1),k=l("",3),M=s("ol",{start:"9"},[s("li",null,[a("三角形の面樍 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"a"),s("msub",null,[s("mi",null,"h"),s("mi",null,"a")]),s("mo",null,"="),s("mi",null,"p"),s("mi",null,"r"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"p"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"a"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"c"),s("mo",{stretchy:"false"},")")])]),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"4"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"=\\frac{1}{2} a h_a=p r=\\sqrt{p(p-a)(p-b)(p-c)}=a b c / 4 R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"h"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as m,c as n,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const K=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 1","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/1.md","filePath":"academic/physics/ipho-formulas-jpn/1.md","lastUpdated":1684859551000}'),c={name:"academic/physics/ipho-formulas-jpn/1.md"},r=l("",3),h=s("ol",null,[s("li",null,[s("p",null,"Taylor 展開（アバウトに切り捨てる:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mo",null,"∑"),s("msup",null,[s("mi",null,"F"),s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mo",{stretchy:"false"},")")])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mi",null,"n")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"F(x)=F\\left(x_{0}\\right)+\\sum F^{(n)}\\left(x_{0}\\right)\\left(x-x_{0}\\right)^{n} / n ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mclose mtight"},")")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8043em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"n")])])])])]),s("p",null,"線形近似（特別な場合）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"F"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"F"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"F(x) \\approx F\\left(x_{0}\\right)+F^{\\prime}\\left(x_{0}\\right)\\left(x-x_{0}\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0519em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8019em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])]),s("p",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"x"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"|x| \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" のときの例 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":")]),s("annotation",{encoding:"application/x-tex"},":")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"x"),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"e"),s("mi",null,"x")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\sin x \\approx x, \\cos x \\approx 1-x^{2} / 2, e^{x} \\approx 1+x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x")])])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ln"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"≈"),s("mi",null,"x"),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"n")]),s("mo",null,"≈"),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"n"),s("mi",null,"x")]),s("annotation",{encoding:"application/x-tex"},"\\ln (1+x) \\approx x,(1+x)^{n} \\approx 1+n x ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"ln"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal"},"x")])])])])])])],-1),o=s("h3",{id:"_1-2-摂動法",tabindex:"-1"},[a("1.2: 摂動法 "),s("a",{class:"header-anchor",href:"#_1-2-摂動法","aria-label":'Permalink to "1.2: 摂動法"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("摂動法：摂動のない（直接解ける）問題の解を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" 番目の近似値として求め，前の似値に基づく次の近似値の補正を繰り返して解を求める．")])],-1),u=s("h3",{id:"_1-3-定数係数線形微分方程式",tabindex:"-1"},[a("1.3: 定数係数線形微分方程式 "),s("a",{class:"header-anchor",href:"#_1-3-定数係数線形微分方程式","aria-label":'Permalink to "1.3: 定数係数線形微分方程式"'},"​")],-1),y=s("ol",{start:"3"},[s("li",null,[s("p",null,[a("定数係数線形微分方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"y"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])]),s("mo",null,"+"),s("mi",null,"b"),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"+"),s("mi",null,"c"),s("mi",null,"y"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a y^{\\prime \\prime}+b y^{\\prime}+c y=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9463em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"cy"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の解:")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"A"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"1")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mo",null,"+"),s("mi",null,"B"),s("mi",null,"exp"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",{fence:"true"},")")]),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"y=A \\exp \\left(\\lambda_1 x\\right)+B \\exp \\left(\\lambda_2 x\\right) \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"exp"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])]),s("p",null,[a("ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" は特性方程式 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("msup",null,[s("mi",null,"λ"),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"λ"),s("mo",null,"+"),s("mi",null,"c"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"a \\lambda^2+b \\lambda+c=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8974em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"bλ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(" の異な る 2 解. もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"b"),s("mo",{separator:"true"},","),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a, b, c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" が実数で特性方程式の解が複素数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"λ"),s("mrow",null,[s("mn",null,"1"),s("mo",{separator:"true"},","),s("mn",null,"2")])]),s("mo",null,"="),s("mi",null,"γ"),s("mo",null,"±"),s("mi",null,"i"),s("mi",null,"ω")]),s("annotation",{encoding:"application/x-tex"},"\\lambda_{1,2}=\\gamma \\pm i \\omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mtight"},"2")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω")])])]),a(" ならば,")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"="),s("mi",null,"C"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"γ"),s("mi",null,"x")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"ω"),s("mi",null,"x"),s("mo",null,"+"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y=C e^{\\gamma x} \\sin \\left(\\omega x+\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"x")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])])])],-1),d=s("h3",{id:"_1-4-複素数",tabindex:"-1"},[a("1.4: 複素数 "),s("a",{class:"header-anchor",href:"#_1-4-複素数","aria-label":'Permalink to "1.4: 複素数"'},"​")],-1),v=s("ol",{start:"4"},[s("li",null,[s("p",null,"複素数"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"z"),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",{separator:"true"},","),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("mi",null,"a"),s("mo",null,"−"),s("mi",null,"b"),s("mi",null,"i"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("msup",null,[s("mi",{mathvariant:"normal"},"∣"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"z"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mfrac",null,[s("mi",null,"b"),s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"z"),s("mi",{mathvariant:"normal"},"∣")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"Re"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"+"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Im"),s("mo",null,"⁡"),s("mi",null,"z"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"z"),s("mo",null,"−"),s("mover",{accent:"true"},[s("mi",null,"z"),s("mo",null,"ˉ")]),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"i")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",{fence:"true"},"∣")]),s("mrow",null,[s("mo",{fence:"true"},"∣"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",{fence:"true"},"∣")]),s("mo",{separator:"true"},","),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")]),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"1")]),s("mo",null,"+"),s("mi",null,"arg"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"z"),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"+"),s("mi",null,"i"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"+"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"φ")])]),s("mo",null,"−"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"i"),s("mi",null,"φ")])])]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"i")])])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} z=a+b i=|z| e^{i \\varphi}, \\bar{z}=a-b i=|z| e^{-i \\varphi} \\\\ |z|^2=z \\bar{z}=a^2+b^2, \\varphi=\\arg z=\\arcsin \\frac{b}{|z|} \\\\ \\operatorname{Re} z=(z+\\bar{z}) / 2, \\operatorname{Im} z=(z-\\bar{z}) / 2 i \\\\ \\left|z_1 z_2\\right|=\\left|z_1\\right|\\left|z_2\\right|, \\arg z_1 z_2=\\arg z_1+\\arg z_2 \\\\ e^{i \\varphi}=\\cos \\varphi+i \\sin \\varphi \\\\ \\cos \\varphi=\\frac{e^{i \\varphi}+e^{-i \\varphi}}{2}, \\sin \\varphi=\\frac{e^{i \\varphi}-e^{-i \\varphi}}{2 i} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.1644em","vertical-align":"-5.3322em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.8322em"}},[s("span",{style:{top:"-8.4592em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"bi"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])]),s("span",{style:{top:"-6.4278em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord"},"∣")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.936em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-4.3518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Re")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"Im")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5678em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1944em"}},[s("span",{class:"mord"},"ˉ")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-2.8518em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},"∣")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.044em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.3171em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8747em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"0.8446em"}},[s("span",{class:"pstrut",style:{height:"3.5017em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5017em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"i")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mord mathnormal mtight"},"φ")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.3322em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_1-5-ベクトルの内積と外積",tabindex:"-1"},[a("1.5: ベクトルの内積と外積 "),s("a",{class:"header-anchor",href:"#_1-5-ベクトルの内積と外積","aria-label":'Permalink to "1.5: ベクトルの内積と外積"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[s("p",null,[a("ベクトルの内積と外積は分配法則が成立する : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"a"),s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"a(b+c)=a b+a c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.0833em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"φ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mo",null,"−"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⊥"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"y")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"y")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",null,"+"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"a"),s("mi",null,"z")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"x")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"b"),s("mi",null,"z")]),s("mo",{fence:"true"},")")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"y")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"−"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"bold-italic"},"c")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{a} \\cdot \\boldsymbol{b}=\\boldsymbol{b} \\cdot \\boldsymbol{a}=a_x b_x+a_y b_y+\\cdots=a b \\cos \\varphi \\\\ |\\boldsymbol{a} \\times \\boldsymbol{b}|=a b \\sin \\varphi, \\boldsymbol{a} \\times \\boldsymbol{b}=-\\boldsymbol{b} \\times \\boldsymbol{a} \\perp \\boldsymbol{a}, \\boldsymbol{b} \\\\ \\boldsymbol{a} \\times \\boldsymbol{b}=\\left(a_y b_z-a_z b_y\\right) \\boldsymbol{e}_x+\\left(a_z b_x-a_x b_z\\right) \\boldsymbol{e}_y+\\cdots \\\\ \\boldsymbol{a} \\times[\\boldsymbol{b} \\times \\boldsymbol{c}]=(\\boldsymbol{a} \\cdot \\boldsymbol{c}) \\boldsymbol{b}-(\\boldsymbol{a} \\cdot \\boldsymbol{b}) \\boldsymbol{c} \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"6em","vertical-align":"-2.75em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.25em"}},[s("span",{style:{top:"-5.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])]),s("span",{style:{top:"-3.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},"⋯")])]),s("span",{style:{top:"-0.91em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},")"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.75em"}},[s("span")])])])])])])])])])])]),s("p",null,"スカラー三重積（3 つのベクトルで張られる平行四面 体の体積）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},")"),s("mo",null,"≡"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"⋅"),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{stretchy:"false"},"]"),s("mo",null,"="),s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{stretchy:"false"},"]"),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"b"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"c"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}) \\equiv \\boldsymbol{a} \\cdot[\\boldsymbol{b} \\times \\boldsymbol{c}]=[\\boldsymbol{a} \\times \\boldsymbol{b}] \\cdot \\boldsymbol{c}=(\\boldsymbol{b}, \\boldsymbol{c}, \\boldsymbol{a}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4445em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mclose"},"]"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"c")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mclose"},")")])])])])])])],-1),z=s("h3",{id:"_1-6-余弦定理と正弦定理",tabindex:"-1"},[a("1.6: 余弦定理と正弦定理 "),s("a",{class:"header-anchor",href:"#_1-6-余弦定理と正弦定理","aria-label":'Permalink to "1.6: 余弦定理と正弦定理"'},"​")],-1),f=s("ol",{start:"6"},[s("li",null,[a("余弦定理と正弦定理："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")]),s("mo",null,"−"),s("mn",null,"2"),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"C")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"a"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"A"),s("mo",null,"="),s("mi",null,"b"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"B"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & c^2=a^2+b^2-2 a b \\cos C \\\\ & a / \\sin A=b / \\sin B=2 R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.7621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.2621em"}},[s("span",{class:"pstrut",style:{height:"2.8641em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"ab"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_1-7-三角法",tabindex:"-1"},[a("1.7: 三角法 "),s("a",{class:"header-anchor",href:"#_1-7-三角法","aria-label":'Permalink to "1.7: 三角法"'},"​")],-1),_=s("ol",{start:"7"},[s("li",null,[s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"±"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"∓"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"β")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"±"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"∓"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"tan"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mi",null,"α"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mn",null,"2"),s("mi",null,"α")]),s("mn",null,"2")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β"),s("mo",{stretchy:"false"},")")]),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β"),s("mo",null,"="),s("mn",null,"2"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"+"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",null,"+"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mfrac",null,[s("mrow",null,[s("mi",null,"α"),s("mo",null,"−"),s("mi",null,"β")]),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",null,"…")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\sin (\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\\\ & \\cos (\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta \\\\ & \\tan (\\alpha \\pm \\beta)=(\\tan \\alpha \\pm \\tan \\beta) /(1 \\mp \\tan \\alpha \\tan \\beta) \\\\ & \\cos ^2 \\alpha=\\frac{1+\\cos 2 \\alpha}{2}, \\sin ^2 \\alpha=\\frac{1-\\cos 2 \\alpha}{2} \\\\ & \\cos \\alpha \\cos \\beta=\\frac{\\cos (\\alpha+\\beta)+\\cos (\\alpha-\\beta)}{2}, \\ldots \\\\ & \\cos \\alpha+\\cos \\beta=2\\left(\\cos \\frac{\\alpha+\\beta}{2}+\\cos \\frac{\\alpha-\\beta}{2}\\right), \\ldots \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.9205em","vertical-align":"-5.7102em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.2102em"}},[s("span",{style:{top:"-8.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-7.3202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])]),s("span",{style:{top:"-5.8202em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"±"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"∓"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"tan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-3.8388em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])]),s("span",{style:{top:"-1.4258em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.427em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mclose"},")")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])]),s("span",{style:{top:"1.0102em"}},[s("span",{class:"pstrut",style:{height:"3.45em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"…")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.7102em"}},[s("span")])])])])])])])])])])])])],-1),k=l("",3),M=s("ol",{start:"9"},[s("li",null,[a("三角形の面樍 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"a"),s("msub",null,[s("mi",null,"h"),s("mi",null,"a")]),s("mo",null,"="),s("mi",null,"p"),s("mi",null,"r"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"p"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"a"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"p"),s("mo",null,"−"),s("mi",null,"c"),s("mo",{stretchy:"false"},")")])]),s("mo",null,"="),s("mi",null,"a"),s("mi",null,"b"),s("mi",null,"c"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"4"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"=\\frac{1}{2} a h_a=p r=\\sqrt{p(p-a)(p-b)(p-c)}=a b c / 4 R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"h"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_10.md.31925895.js b/assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_10.md.31925895.js rename to assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.js index 4ec24e4c..42d50b92 100644 --- a/assets/academic_physics_ipho-formulas-jpn_10.md.31925895.js +++ b/assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.js @@ -1,4 +1,4 @@ -import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const is=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 10","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/10.md","filePath":"academic/physics/ipho-formulas-jpn/10.md","lastUpdated":1684858945000}'),m={name:"academic/physics/ipho-formulas-jpn/10.md"},e=s("h1",{id:"formulas-for-ipho-日本語版-section-10",tabindex:"-1"},[a("Formulas for IPhO 日本語版: Section 10 "),s("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-10","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 10"'},"​")],-1),i=s("h2",{id:"_10-熱力学",tabindex:"-1"},[a("10: 熱力学 "),s("a",{class:"header-anchor",href:"#_10-熱力学","aria-label":'Permalink to "10: 熱力学"'},"​")],-1),r=s("h3",{id:"_10-1",tabindex:"-1"},[a("10.1: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(),s("a",{class:"header-anchor",href:"#_10-1","aria-label":'Permalink to "10.1: $p V=\\frac{w}{M} R T$"'},"​")],-1),p=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(".")])],-1),c=s("h3",{id:"_10-2-モルの気体の内部エネルギー",tabindex:"-1"},[a("10.2: モルの気体の内部エネルギー "),s("a",{class:"header-anchor",href:"#_10-2-モルの気体の内部エネルギー","aria-label":'Permalink to "10.2: モルの気体の内部エネルギー"'},"​")],-1),h=s("ol",{start:"2"},[s("li",null,[a("1 モルの気体の内部エネルギー: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"i"),s("mn",null,"2")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"U=\\frac{i}{2} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2007em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8557em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(" [訳者注: 単 原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"3")]),s("annotation",{encoding:"application/x-tex"},"i=3")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"3")])])]),a(", 二原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"5"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"i=5]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"5"),s("span",{class:"mclose"},"]")])])]),a(".")])],-1),o=s("h3",{id:"_10-3-標準状態",tabindex:"-1"},[a("10.3: 標準状態 "),s("a",{class:"header-anchor",href:"#_10-3-標準状態","aria-label":'Permalink to "10.3: 標準状態"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("標準状態での 1 モルの気体の体積は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"22.4"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")])]),s("annotation",{encoding:"application/x-tex"},"22.4 \\mathrm{~L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"22.4"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")])])])]),a(".")])],-1),u=s("h3",{id:"_10-4-断熱過程",tabindex:"-1"},[a("10.4: 断熱過程 "),s("a",{class:"header-anchor",href:"#_10-4-断熱過程","aria-label":'Permalink to "10.4: 断熱過程"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("断熱過程: 音速に比べて遅く, 熱の出入りがない. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("msup",null,[s("mi",null,"V"),s("mi",null,"γ")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"p V^\\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"T"),s("msup",null,[s("mi",null,"V"),s("mrow",null,[s("mi",null,"γ"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left(T V^{\\gamma-1}=\\right.")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mclose nulldelimiter"})])])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])],-1),y=s("h3",{id:"_10-5-γ-cp-cv-i-2-i",tabindex:"-1"},[a("10.5: γ=Cp/Cv=(i+2)/i "),s("a",{class:"header-anchor",href:"#_10-5-γ-cp-cv-i-2-i","aria-label":'Permalink to "10.5: γ=Cp/Cv=(i+2)/i"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"γ"),s("mo",null,"="),s("msub",null,[s("mi",null,"c"),s("mi",null,"p")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"c"),s("mi",null,"v")]),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i")]),s("annotation",{encoding:"application/x-tex"},"\\gamma=c_p / c_v=(i+2) / i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0361em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"p")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"v")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"i")])])]),a(".")])],-1),x=s("h3",{id:"_10-6-boltzmann-分布",tabindex:"-1"},[a("10.6: Boltzmann 分布 "),s("a",{class:"header-anchor",href:"#_10-6-boltzmann-分布","aria-label":'Permalink to "10.6: Boltzmann 分布"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Boltzmann 分布 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ"),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"M"),s("mi",null,"g"),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",null,"T")])]),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"U"),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\rho=\\rho_0 e^{-M g h / R T}=\\rho_0 e^{-U / k_B T} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord mathnormal mtight"},"h"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"RT")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),w=s("h3",{id:"_10-7-maxwell-分布",tabindex:"-1"},[a("10.7: Maxwell 分布 "),s("a",{class:"header-anchor",href:"#_10-7-maxwell-分布","aria-label":'Permalink to "10.7: Maxwell 分布"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("Maxwell 分布（v の速さをもつ分子の数）"),s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"訳者注"),s("p",null,[a("位相空間で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}+\\mathrm{d} \\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" の間にある分子の数の分布 であり，v の速さをもつ分子の数の分布とは異なる] "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∝"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"m"),s("msup",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\propto e^{-m \\boldsymbol{v}^2 / 2 k_B T}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9869em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9869em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"m"),s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord boldsymbol mtight",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8913em"}},[s("span",{style:{top:"-2.931em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mtight"},"/2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),_=s("h3",{id:"_10-8-大気圧",tabindex:"-1"},[a("10.8: 大気圧 "),s("a",{class:"header-anchor",href:"#_10-8-大気圧","aria-label":'Permalink to "10.8: 大気圧"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("大気圧 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"≪"),s("mi",null,"p")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\ll p")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"="),s("mi",null,"ρ"),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p=\\rho g \\Delta h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"h")])])]),a(".")])],-1),M=s("h3",{id:"_10-9-公式",tabindex:"-1"},[a("10.9: 公式 "),s("a",{class:"header-anchor",href:"#_10-9-公式","aria-label":'Permalink to "10.9: 公式"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"3")]),s("mi",null,"m"),s("mi",null,"n"),s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",null,"="),s("mi",null,"n"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T"),s("mo",{stretchy:"false"},"("),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"p=\\frac{1}{3} m n \\overline{v^2}=n k_B T(n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2851em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"mn"),s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"n")])])]),a(" は数密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msqrt",null,[s("mover",{accent:"true"},[s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",{stretchy:"true"},"‾")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"), \\sqrt{\\overline{\\overline{v^2}}}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.84em","vertical-align":"-0.2849em"}}),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5551em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-4.0601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-3.5151em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const is=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 10","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/10.md","filePath":"academic/physics/ipho-formulas-jpn/10.md","lastUpdated":1684859551000}'),m={name:"academic/physics/ipho-formulas-jpn/10.md"},e=s("h1",{id:"formulas-for-ipho-日本語版-section-10",tabindex:"-1"},[a("Formulas for IPhO 日本語版: Section 10 "),s("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-10","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 10"'},"​")],-1),i=s("h2",{id:"_10-熱力学",tabindex:"-1"},[a("10: 熱力学 "),s("a",{class:"header-anchor",href:"#_10-熱力学","aria-label":'Permalink to "10: 熱力学"'},"​")],-1),r=s("h3",{id:"_10-1",tabindex:"-1"},[a("10.1: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(),s("a",{class:"header-anchor",href:"#_10-1","aria-label":'Permalink to "10.1: $p V=\\frac{w}{M} R T$"'},"​")],-1),p=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(".")])],-1),c=s("h3",{id:"_10-2-モルの気体の内部エネルギー",tabindex:"-1"},[a("10.2: モルの気体の内部エネルギー "),s("a",{class:"header-anchor",href:"#_10-2-モルの気体の内部エネルギー","aria-label":'Permalink to "10.2: モルの気体の内部エネルギー"'},"​")],-1),h=s("ol",{start:"2"},[s("li",null,[a("1 モルの気体の内部エネルギー: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"i"),s("mn",null,"2")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"U=\\frac{i}{2} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2007em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8557em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(" [訳者注: 単 原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"3")]),s("annotation",{encoding:"application/x-tex"},"i=3")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"3")])])]),a(", 二原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"5"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"i=5]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"5"),s("span",{class:"mclose"},"]")])])]),a(".")])],-1),o=s("h3",{id:"_10-3-標準状態",tabindex:"-1"},[a("10.3: 標準状態 "),s("a",{class:"header-anchor",href:"#_10-3-標準状態","aria-label":'Permalink to "10.3: 標準状態"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("標準状態での 1 モルの気体の体積は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"22.4"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")])]),s("annotation",{encoding:"application/x-tex"},"22.4 \\mathrm{~L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"22.4"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")])])])]),a(".")])],-1),u=s("h3",{id:"_10-4-断熱過程",tabindex:"-1"},[a("10.4: 断熱過程 "),s("a",{class:"header-anchor",href:"#_10-4-断熱過程","aria-label":'Permalink to "10.4: 断熱過程"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("断熱過程: 音速に比べて遅く, 熱の出入りがない. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("msup",null,[s("mi",null,"V"),s("mi",null,"γ")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"p V^\\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"T"),s("msup",null,[s("mi",null,"V"),s("mrow",null,[s("mi",null,"γ"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left(T V^{\\gamma-1}=\\right.")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mclose nulldelimiter"})])])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])],-1),y=s("h3",{id:"_10-5-γ-cp-cv-i-2-i",tabindex:"-1"},[a("10.5: γ=Cp/Cv=(i+2)/i "),s("a",{class:"header-anchor",href:"#_10-5-γ-cp-cv-i-2-i","aria-label":'Permalink to "10.5: γ=Cp/Cv=(i+2)/i"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"γ"),s("mo",null,"="),s("msub",null,[s("mi",null,"c"),s("mi",null,"p")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"c"),s("mi",null,"v")]),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i")]),s("annotation",{encoding:"application/x-tex"},"\\gamma=c_p / c_v=(i+2) / i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0361em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"p")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"v")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"i")])])]),a(".")])],-1),x=s("h3",{id:"_10-6-boltzmann-分布",tabindex:"-1"},[a("10.6: Boltzmann 分布 "),s("a",{class:"header-anchor",href:"#_10-6-boltzmann-分布","aria-label":'Permalink to "10.6: Boltzmann 分布"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Boltzmann 分布 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ"),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"M"),s("mi",null,"g"),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",null,"T")])]),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"U"),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\rho=\\rho_0 e^{-M g h / R T}=\\rho_0 e^{-U / k_B T} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord mathnormal mtight"},"h"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"RT")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),w=s("h3",{id:"_10-7-maxwell-分布",tabindex:"-1"},[a("10.7: Maxwell 分布 "),s("a",{class:"header-anchor",href:"#_10-7-maxwell-分布","aria-label":'Permalink to "10.7: Maxwell 分布"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("Maxwell 分布（v の速さをもつ分子の数）"),s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"訳者注"),s("p",null,[a("位相空間で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}+\\mathrm{d} \\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" の間にある分子の数の分布 であり，v の速さをもつ分子の数の分布とは異なる] "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∝"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"m"),s("msup",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\propto e^{-m \\boldsymbol{v}^2 / 2 k_B T}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9869em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9869em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"m"),s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord boldsymbol mtight",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8913em"}},[s("span",{style:{top:"-2.931em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mtight"},"/2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),_=s("h3",{id:"_10-8-大気圧",tabindex:"-1"},[a("10.8: 大気圧 "),s("a",{class:"header-anchor",href:"#_10-8-大気圧","aria-label":'Permalink to "10.8: 大気圧"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("大気圧 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"≪"),s("mi",null,"p")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\ll p")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"="),s("mi",null,"ρ"),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p=\\rho g \\Delta h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"h")])])]),a(".")])],-1),M=s("h3",{id:"_10-9-公式",tabindex:"-1"},[a("10.9: 公式 "),s("a",{class:"header-anchor",href:"#_10-9-公式","aria-label":'Permalink to "10.9: 公式"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"3")]),s("mi",null,"m"),s("mi",null,"n"),s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",null,"="),s("mi",null,"n"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T"),s("mo",{stretchy:"false"},"("),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"p=\\frac{1}{3} m n \\overline{v^2}=n k_B T(n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2851em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"mn"),s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"n")])])]),a(" は数密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msqrt",null,[s("mover",{accent:"true"},[s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",{stretchy:"true"},"‾")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"), \\sqrt{\\overline{\\overline{v^2}}}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.84em","vertical-align":"-0.2849em"}}),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5551em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-4.0601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-3.5151em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_10.md.31925895.lean.js b/assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_10.md.31925895.lean.js rename to assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.lean.js index 4ec24e4c..42d50b92 100644 --- a/assets/academic_physics_ipho-formulas-jpn_10.md.31925895.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_10.md.53acc7b0.lean.js @@ -1,4 +1,4 @@ -import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const is=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 10","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/10.md","filePath":"academic/physics/ipho-formulas-jpn/10.md","lastUpdated":1684858945000}'),m={name:"academic/physics/ipho-formulas-jpn/10.md"},e=s("h1",{id:"formulas-for-ipho-日本語版-section-10",tabindex:"-1"},[a("Formulas for IPhO 日本語版: Section 10 "),s("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-10","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 10"'},"​")],-1),i=s("h2",{id:"_10-熱力学",tabindex:"-1"},[a("10: 熱力学 "),s("a",{class:"header-anchor",href:"#_10-熱力学","aria-label":'Permalink to "10: 熱力学"'},"​")],-1),r=s("h3",{id:"_10-1",tabindex:"-1"},[a("10.1: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(),s("a",{class:"header-anchor",href:"#_10-1","aria-label":'Permalink to "10.1: $p V=\\frac{w}{M} R T$"'},"​")],-1),p=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(".")])],-1),c=s("h3",{id:"_10-2-モルの気体の内部エネルギー",tabindex:"-1"},[a("10.2: モルの気体の内部エネルギー "),s("a",{class:"header-anchor",href:"#_10-2-モルの気体の内部エネルギー","aria-label":'Permalink to "10.2: モルの気体の内部エネルギー"'},"​")],-1),h=s("ol",{start:"2"},[s("li",null,[a("1 モルの気体の内部エネルギー: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"i"),s("mn",null,"2")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"U=\\frac{i}{2} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2007em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8557em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(" [訳者注: 単 原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"3")]),s("annotation",{encoding:"application/x-tex"},"i=3")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"3")])])]),a(", 二原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"5"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"i=5]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"5"),s("span",{class:"mclose"},"]")])])]),a(".")])],-1),o=s("h3",{id:"_10-3-標準状態",tabindex:"-1"},[a("10.3: 標準状態 "),s("a",{class:"header-anchor",href:"#_10-3-標準状態","aria-label":'Permalink to "10.3: 標準状態"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("標準状態での 1 モルの気体の体積は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"22.4"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")])]),s("annotation",{encoding:"application/x-tex"},"22.4 \\mathrm{~L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"22.4"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")])])])]),a(".")])],-1),u=s("h3",{id:"_10-4-断熱過程",tabindex:"-1"},[a("10.4: 断熱過程 "),s("a",{class:"header-anchor",href:"#_10-4-断熱過程","aria-label":'Permalink to "10.4: 断熱過程"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("断熱過程: 音速に比べて遅く, 熱の出入りがない. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("msup",null,[s("mi",null,"V"),s("mi",null,"γ")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"p V^\\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"T"),s("msup",null,[s("mi",null,"V"),s("mrow",null,[s("mi",null,"γ"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left(T V^{\\gamma-1}=\\right.")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mclose nulldelimiter"})])])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])],-1),y=s("h3",{id:"_10-5-γ-cp-cv-i-2-i",tabindex:"-1"},[a("10.5: γ=Cp/Cv=(i+2)/i "),s("a",{class:"header-anchor",href:"#_10-5-γ-cp-cv-i-2-i","aria-label":'Permalink to "10.5: γ=Cp/Cv=(i+2)/i"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"γ"),s("mo",null,"="),s("msub",null,[s("mi",null,"c"),s("mi",null,"p")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"c"),s("mi",null,"v")]),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i")]),s("annotation",{encoding:"application/x-tex"},"\\gamma=c_p / c_v=(i+2) / i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0361em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"p")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"v")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"i")])])]),a(".")])],-1),x=s("h3",{id:"_10-6-boltzmann-分布",tabindex:"-1"},[a("10.6: Boltzmann 分布 "),s("a",{class:"header-anchor",href:"#_10-6-boltzmann-分布","aria-label":'Permalink to "10.6: Boltzmann 分布"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Boltzmann 分布 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ"),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"M"),s("mi",null,"g"),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",null,"T")])]),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"U"),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\rho=\\rho_0 e^{-M g h / R T}=\\rho_0 e^{-U / k_B T} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord mathnormal mtight"},"h"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"RT")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),w=s("h3",{id:"_10-7-maxwell-分布",tabindex:"-1"},[a("10.7: Maxwell 分布 "),s("a",{class:"header-anchor",href:"#_10-7-maxwell-分布","aria-label":'Permalink to "10.7: Maxwell 分布"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("Maxwell 分布（v の速さをもつ分子の数）"),s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"訳者注"),s("p",null,[a("位相空間で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}+\\mathrm{d} \\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" の間にある分子の数の分布 であり，v の速さをもつ分子の数の分布とは異なる] "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∝"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"m"),s("msup",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\propto e^{-m \\boldsymbol{v}^2 / 2 k_B T}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9869em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9869em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"m"),s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord boldsymbol mtight",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8913em"}},[s("span",{style:{top:"-2.931em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mtight"},"/2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),_=s("h3",{id:"_10-8-大気圧",tabindex:"-1"},[a("10.8: 大気圧 "),s("a",{class:"header-anchor",href:"#_10-8-大気圧","aria-label":'Permalink to "10.8: 大気圧"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("大気圧 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"≪"),s("mi",null,"p")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\ll p")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"="),s("mi",null,"ρ"),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p=\\rho g \\Delta h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"h")])])]),a(".")])],-1),M=s("h3",{id:"_10-9-公式",tabindex:"-1"},[a("10.9: 公式 "),s("a",{class:"header-anchor",href:"#_10-9-公式","aria-label":'Permalink to "10.9: 公式"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"3")]),s("mi",null,"m"),s("mi",null,"n"),s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",null,"="),s("mi",null,"n"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T"),s("mo",{stretchy:"false"},"("),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"p=\\frac{1}{3} m n \\overline{v^2}=n k_B T(n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2851em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"mn"),s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"n")])])]),a(" は数密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msqrt",null,[s("mover",{accent:"true"},[s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",{stretchy:"true"},"‾")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"), \\sqrt{\\overline{\\overline{v^2}}}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.84em","vertical-align":"-0.2849em"}}),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5551em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-4.0601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-3.5151em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as l,c as t,o as n,z as s,a}from"./chunks/framework.a1bac067.js";const is=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 10","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/10.md","filePath":"academic/physics/ipho-formulas-jpn/10.md","lastUpdated":1684859551000}'),m={name:"academic/physics/ipho-formulas-jpn/10.md"},e=s("h1",{id:"formulas-for-ipho-日本語版-section-10",tabindex:"-1"},[a("Formulas for IPhO 日本語版: Section 10 "),s("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-10","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 10"'},"​")],-1),i=s("h2",{id:"_10-熱力学",tabindex:"-1"},[a("10: 熱力学 "),s("a",{class:"header-anchor",href:"#_10-熱力学","aria-label":'Permalink to "10: 熱力学"'},"​")],-1),r=s("h3",{id:"_10-1",tabindex:"-1"},[a("10.1: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(),s("a",{class:"header-anchor",href:"#_10-1","aria-label":'Permalink to "10.1: $p V=\\frac{w}{M} R T$"'},"​")],-1),p=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mi",null,"V"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"w"),s("mi",null,"M")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"p V=\\frac{w}{M} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02691em"}},"w")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(".")])],-1),c=s("h3",{id:"_10-2-モルの気体の内部エネルギー",tabindex:"-1"},[a("10.2: モルの気体の内部エネルギー "),s("a",{class:"header-anchor",href:"#_10-2-モルの気体の内部エネルギー","aria-label":'Permalink to "10.2: モルの気体の内部エネルギー"'},"​")],-1),h=s("ol",{start:"2"},[s("li",null,[a("1 モルの気体の内部エネルギー: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"i"),s("mn",null,"2")]),s("mi",null,"R"),s("mi",null,"T")]),s("annotation",{encoding:"application/x-tex"},"U=\\frac{i}{2} R T")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2007em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8557em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"RT")])])]),a(" [訳者注: 単 原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"3")]),s("annotation",{encoding:"application/x-tex"},"i=3")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"3")])])]),a(", 二原子分子理想気体 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"5"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"i=5]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6595em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"5"),s("span",{class:"mclose"},"]")])])]),a(".")])],-1),o=s("h3",{id:"_10-3-標準状態",tabindex:"-1"},[a("10.3: 標準状態 "),s("a",{class:"header-anchor",href:"#_10-3-標準状態","aria-label":'Permalink to "10.3: 標準状態"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("標準状態での 1 モルの気体の体積は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"22.4"),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"L")])]),s("annotation",{encoding:"application/x-tex"},"22.4 \\mathrm{~L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"22.4"),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"L")])])])]),a(".")])],-1),u=s("h3",{id:"_10-4-断熱過程",tabindex:"-1"},[a("10.4: 断熱過程 "),s("a",{class:"header-anchor",href:"#_10-4-断熱過程","aria-label":'Permalink to "10.4: 断熱過程"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("断熱過程: 音速に比べて遅く, 熱の出入りがない. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("msup",null,[s("mi",null,"V"),s("mi",null,"γ")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"p V^\\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"T"),s("msup",null,[s("mi",null,"V"),s("mrow",null,[s("mi",null,"γ"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left(T V^{\\gamma-1}=\\right.")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mclose nulldelimiter"})])])])]),a(" const. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])],-1),y=s("h3",{id:"_10-5-γ-cp-cv-i-2-i",tabindex:"-1"},[a("10.5: γ=Cp/Cv=(i+2)/i "),s("a",{class:"header-anchor",href:"#_10-5-γ-cp-cv-i-2-i","aria-label":'Permalink to "10.5: γ=Cp/Cv=(i+2)/i"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"γ"),s("mo",null,"="),s("msub",null,[s("mi",null,"c"),s("mi",null,"p")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"c"),s("mi",null,"v")]),s("mo",null,"="),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i")]),s("annotation",{encoding:"application/x-tex"},"\\gamma=c_p / c_v=(i+2) / i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0361em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"p")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"v")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"i")])])]),a(".")])],-1),x=s("h3",{id:"_10-6-boltzmann-分布",tabindex:"-1"},[a("10.6: Boltzmann 分布 "),s("a",{class:"header-anchor",href:"#_10-6-boltzmann-分布","aria-label":'Permalink to "10.6: Boltzmann 分布"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Boltzmann 分布 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ"),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"M"),s("mi",null,"g"),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",null,"T")])]),s("mo",null,"="),s("msub",null,[s("mi",null,"ρ"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"U"),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\rho=\\rho_0 e^{-M g h / R T}=\\rho_0 e^{-U / k_B T} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord mathnormal mtight"},"h"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"RT")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1324em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.938em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mord mtight"},"/"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),w=s("h3",{id:"_10-7-maxwell-分布",tabindex:"-1"},[a("10.7: Maxwell 分布 "),s("a",{class:"header-anchor",href:"#_10-7-maxwell-分布","aria-label":'Permalink to "10.7: Maxwell 分布"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("Maxwell 分布（v の速さをもつ分子の数）"),s("div",{class:"tip custom-block"},[s("p",{class:"custom-block-title"},"訳者注"),s("p",null,[a("位相空間で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"+"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}+\\mathrm{d} \\boldsymbol{v}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),a(" の間にある分子の数の分布 であり，v の速さをもつ分子の数の分布とは異なる] "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∝"),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"m"),s("msup",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T")])])]),s("annotation",{encoding:"application/x-tex"},"\\propto e^{-m \\boldsymbol{v}^2 / 2 k_B T}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9869em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9869em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight"},"m"),s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord boldsymbol mtight",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8913em"}},[s("span",{style:{top:"-2.931em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mtight"},"/2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.3567em","margin-left":"-0.0315em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1433em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.13889em"}},"T")])])])])])])])])])])])])])])],-1),_=s("h3",{id:"_10-8-大気圧",tabindex:"-1"},[a("10.8: 大気圧 "),s("a",{class:"header-anchor",href:"#_10-8-大気圧","aria-label":'Permalink to "10.8: 大気圧"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("大気圧 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"≪"),s("mi",null,"p")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\ll p")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mo",null,"="),s("mi",null,"ρ"),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"\\Delta p=\\rho g \\Delta h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"h")])])]),a(".")])],-1),M=s("h3",{id:"_10-9-公式",tabindex:"-1"},[a("10.9: 公式 "),s("a",{class:"header-anchor",href:"#_10-9-公式","aria-label":'Permalink to "10.9: 公式"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"p"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"3")]),s("mi",null,"m"),s("mi",null,"n"),s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",null,"="),s("mi",null,"n"),s("msub",null,[s("mi",null,"k"),s("mi",null,"B")]),s("mi",null,"T"),s("mo",{stretchy:"false"},"("),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"p=\\frac{1}{3} m n \\overline{v^2}=n k_B T(n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2851em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord mathnormal"},"mn"),s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"n")])])]),a(" は数密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msqrt",null,[s("mover",{accent:"true"},[s("mover",{accent:"true"},[s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",{stretchy:"true"},"‾")]),s("mo",{stretchy:"true"},"‾")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"), \\sqrt{\\overline{\\overline{v^2}}}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.84em","vertical-align":"-0.2849em"}}),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5551em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord overline"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9401em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.8601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-4.0601em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"overline-line",style:{"border-bottom-width":"0.04em"}})])])])])])])]),s("span",{style:{top:"-3.5151em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.js b/assets/academic_physics_ipho-formulas-jpn_11.md.80185913.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.js rename to assets/academic_physics_ipho-formulas-jpn_11.md.80185913.js index ea7d5c5a..175501db 100644 --- a/assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.js +++ b/assets/academic_physics_ipho-formulas-jpn_11.md.80185913.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 11","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/11.md","filePath":"academic/physics/ipho-formulas-jpn/11.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/11.md"},i=l('

Formulas for IPhO 日本語版: Section 11 ​

11: 量子力学 ​

11.1:p=hk ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",{mathvariant:"bold-italic"},"k"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"p"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",null,"ω"),s("mo",null,"="),s("mi",null,"h"),s("mi",null,"ν")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}=\\hbar \\boldsymbol{k}(|\\boldsymbol{p}|=h / \\lambda), E=\\hbar \\omega=h \\nu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6389em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν")])])]),a(".")])],-1),p=l('

11.2: 干渉 ​

  1. 干渉 : 波動光学のように.

11.3: 不確定性 ​

',3),c=s("ol",{start:"3"},[s("li",null,[s("p",null,"不確定性（数学の定理）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\Delta x \\geq \\frac{\\hbar}{2}, \\Delta E \\Delta t \\geq \\frac{\\hbar}{2}, \\Delta \\omega \\Delta t \\geq \\frac{1}{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0074em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])]),s("p",null,[a("滑らかでない場合の定性的な推定には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" の方が適する "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"(\\Delta p \\Delta x \\approx h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" など "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])])],-1),h=s("h3",{id:"_11-4-スペクトル",tabindex:"-1"},[a("11.4: スペクトル "),s("a",{class:"header-anchor",href:"#_11-4-スペクトル","aria-label":'Permalink to "11.4: スペクトル"'},"​")],-1),o=s("ol",{start:"4"},[s("li",null,[a("スペクトル : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h"),s("mi",null,"ν"),s("mo",null,"="),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"E"),s("mi",null,"m")])]),s("annotation",{encoding:"application/x-tex"},"h \\nu=E_n-E_m")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"m")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". スペクトル線の幅は寿 命に関係し, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ")])])]),a(".")])],-1),g=s("h3",{id:"_11-5-振動子",tabindex:"-1"},[a("11.5: 振動子 "),s("a",{class:"header-anchor",href:"#_11-5-振動子","aria-label":'Permalink to "11.5: 振動子"'},"​")],-1),u=s("ol",{start:"5"},[s("li",null,[a("振動子（例えば分子）のエネルギー準位（固有振動数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",null,"h"),s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")])]),s("annotation",{encoding:"application/x-tex"},"\\left.\\nu_0\\right): E_n=\\left(n+\\frac{1}{2}\\right) h \\nu_0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 多数の固有振動数の場合, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"∑"),s("mi",null,"h"),s("msub",null,[s("mi",null,"n"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"ν"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"E=\\sum h n_i \\nu_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(".")])],-1),d=s("h3",{id:"_11-6-トンネル効果",tabindex:"-1"},[a("11.6: トンネル効果 "),s("a",{class:"header-anchor",href:"#_11-6-トンネル効果","aria-label":'Permalink to "11.6: トンネル効果"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("トンネル効果: 幅 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" の障壁 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ")])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ"),s("mo",{stretchy:"false"},"("),s("mi",null,"τ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar(\\tau=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m")])]),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"l / \\sqrt{\\Gamma / m})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mord"},"/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"Γ/"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 11","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/11.md","filePath":"academic/physics/ipho-formulas-jpn/11.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/11.md"},i=l('

Formulas for IPhO 日本語版: Section 11 ​

11: 量子力学 ​

11.1:p=hk ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",{mathvariant:"bold-italic"},"k"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"p"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",null,"ω"),s("mo",null,"="),s("mi",null,"h"),s("mi",null,"ν")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}=\\hbar \\boldsymbol{k}(|\\boldsymbol{p}|=h / \\lambda), E=\\hbar \\omega=h \\nu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6389em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν")])])]),a(".")])],-1),p=l('

11.2: 干渉 ​

  1. 干渉 : 波動光学のように.

11.3: 不確定性 ​

',3),c=s("ol",{start:"3"},[s("li",null,[s("p",null,"不確定性（数学の定理）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\Delta x \\geq \\frac{\\hbar}{2}, \\Delta E \\Delta t \\geq \\frac{\\hbar}{2}, \\Delta \\omega \\Delta t \\geq \\frac{1}{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0074em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])]),s("p",null,[a("滑らかでない場合の定性的な推定には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" の方が適する "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"(\\Delta p \\Delta x \\approx h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" など "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])])],-1),h=s("h3",{id:"_11-4-スペクトル",tabindex:"-1"},[a("11.4: スペクトル "),s("a",{class:"header-anchor",href:"#_11-4-スペクトル","aria-label":'Permalink to "11.4: スペクトル"'},"​")],-1),o=s("ol",{start:"4"},[s("li",null,[a("スペクトル : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h"),s("mi",null,"ν"),s("mo",null,"="),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"E"),s("mi",null,"m")])]),s("annotation",{encoding:"application/x-tex"},"h \\nu=E_n-E_m")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"m")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". スペクトル線の幅は寿 命に関係し, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ")])])]),a(".")])],-1),g=s("h3",{id:"_11-5-振動子",tabindex:"-1"},[a("11.5: 振動子 "),s("a",{class:"header-anchor",href:"#_11-5-振動子","aria-label":'Permalink to "11.5: 振動子"'},"​")],-1),u=s("ol",{start:"5"},[s("li",null,[a("振動子（例えば分子）のエネルギー準位（固有振動数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",null,"h"),s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")])]),s("annotation",{encoding:"application/x-tex"},"\\left.\\nu_0\\right): E_n=\\left(n+\\frac{1}{2}\\right) h \\nu_0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 多数の固有振動数の場合, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"∑"),s("mi",null,"h"),s("msub",null,[s("mi",null,"n"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"ν"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"E=\\sum h n_i \\nu_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(".")])],-1),d=s("h3",{id:"_11-6-トンネル効果",tabindex:"-1"},[a("11.6: トンネル効果 "),s("a",{class:"header-anchor",href:"#_11-6-トンネル効果","aria-label":'Permalink to "11.6: トンネル効果"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("トンネル効果: 幅 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" の障壁 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ")])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ"),s("mo",{stretchy:"false"},"("),s("mi",null,"τ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar(\\tau=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m")])]),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"l / \\sqrt{\\Gamma / m})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mord"},"/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"Γ/"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.lean.js b/assets/academic_physics_ipho-formulas-jpn_11.md.80185913.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.lean.js rename to assets/academic_physics_ipho-formulas-jpn_11.md.80185913.lean.js index 1170f68f..aa8d548f 100644 --- a/assets/academic_physics_ipho-formulas-jpn_11.md.3d1ed58e.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_11.md.80185913.lean.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 11","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/11.md","filePath":"academic/physics/ipho-formulas-jpn/11.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/11.md"},i=l("",3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",{mathvariant:"bold-italic"},"k"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"p"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",null,"ω"),s("mo",null,"="),s("mi",null,"h"),s("mi",null,"ν")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}=\\hbar \\boldsymbol{k}(|\\boldsymbol{p}|=h / \\lambda), E=\\hbar \\omega=h \\nu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6389em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν")])])]),a(".")])],-1),p=l("",3),c=s("ol",{start:"3"},[s("li",null,[s("p",null,"不確定性（数学の定理）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\Delta x \\geq \\frac{\\hbar}{2}, \\Delta E \\Delta t \\geq \\frac{\\hbar}{2}, \\Delta \\omega \\Delta t \\geq \\frac{1}{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0074em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])]),s("p",null,[a("滑らかでない場合の定性的な推定には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" の方が適する "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"(\\Delta p \\Delta x \\approx h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" など "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])])],-1),h=s("h3",{id:"_11-4-スペクトル",tabindex:"-1"},[a("11.4: スペクトル "),s("a",{class:"header-anchor",href:"#_11-4-スペクトル","aria-label":'Permalink to "11.4: スペクトル"'},"​")],-1),o=s("ol",{start:"4"},[s("li",null,[a("スペクトル : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h"),s("mi",null,"ν"),s("mo",null,"="),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"E"),s("mi",null,"m")])]),s("annotation",{encoding:"application/x-tex"},"h \\nu=E_n-E_m")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"m")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". スペクトル線の幅は寿 命に関係し, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ")])])]),a(".")])],-1),g=s("h3",{id:"_11-5-振動子",tabindex:"-1"},[a("11.5: 振動子 "),s("a",{class:"header-anchor",href:"#_11-5-振動子","aria-label":'Permalink to "11.5: 振動子"'},"​")],-1),u=s("ol",{start:"5"},[s("li",null,[a("振動子（例えば分子）のエネルギー準位（固有振動数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",null,"h"),s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")])]),s("annotation",{encoding:"application/x-tex"},"\\left.\\nu_0\\right): E_n=\\left(n+\\frac{1}{2}\\right) h \\nu_0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 多数の固有振動数の場合, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"∑"),s("mi",null,"h"),s("msub",null,[s("mi",null,"n"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"ν"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"E=\\sum h n_i \\nu_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(".")])],-1),d=s("h3",{id:"_11-6-トンネル効果",tabindex:"-1"},[a("11.6: トンネル効果 "),s("a",{class:"header-anchor",href:"#_11-6-トンネル効果","aria-label":'Permalink to "11.6: トンネル効果"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("トンネル効果: 幅 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" の障壁 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ")])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ"),s("mo",{stretchy:"false"},"("),s("mi",null,"τ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar(\\tau=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m")])]),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"l / \\sqrt{\\Gamma / m})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mord"},"/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"Γ/"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 11","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/11.md","filePath":"academic/physics/ipho-formulas-jpn/11.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/11.md"},i=l("",3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",{mathvariant:"bold-italic"},"k"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"p"),s("mi",{mathvariant:"normal"},"∣"),s("mo",null,"="),s("mi",null,"h"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"ℏ"),s("mi",null,"ω"),s("mo",null,"="),s("mi",null,"h"),s("mi",null,"ν")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}=\\hbar \\boldsymbol{k}(|\\boldsymbol{p}|=h / \\lambda), E=\\hbar \\omega=h \\nu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6389em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν")])])]),a(".")])],-1),p=l("",3),c=s("ol",{start:"3"},[s("li",null,[s("p",null,"不確定性（数学の定理）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mi",{mathvariant:"normal"},"ℏ"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"t"),s("mo",null,"≥"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\Delta p \\Delta x \\geq \\frac{\\hbar}{2}, \\Delta E \\Delta t \\geq \\frac{\\hbar}{2}, \\Delta \\omega \\Delta t \\geq \\frac{1}{2} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0519em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3659em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"ℏ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0074em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])]),s("p",null,[a("滑らかでない場合の定性的な推定には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" の方が適する "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"p"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"x"),s("mo",null,"≈"),s("mi",null,"h")]),s("annotation",{encoding:"application/x-tex"},"(\\Delta p \\Delta x \\approx h")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"p"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h")])])]),a(" など "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])]),a(".")])])],-1),h=s("h3",{id:"_11-4-スペクトル",tabindex:"-1"},[a("11.4: スペクトル "),s("a",{class:"header-anchor",href:"#_11-4-スペクトル","aria-label":'Permalink to "11.4: スペクトル"'},"​")],-1),o=s("ol",{start:"4"},[s("li",null,[a("スペクトル : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"h"),s("mi",null,"ν"),s("mo",null,"="),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"E"),s("mi",null,"m")])]),s("annotation",{encoding:"application/x-tex"},"h \\nu=E_n-E_m")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"m")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". スペクトル線の幅は寿 命に関係し, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord"},"ℏ")])])]),a(".")])],-1),g=s("h3",{id:"_11-5-振動子",tabindex:"-1"},[a("11.5: 振動子 "),s("a",{class:"header-anchor",href:"#_11-5-振動子","aria-label":'Permalink to "11.5: 振動子"'},"​")],-1),u=s("ol",{start:"5"},[s("li",null,[a("振動子（例えば分子）のエネルギー準位（固有振動数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("msub",null,[s("mi",null,"E"),s("mi",null,"n")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",null,"h"),s("msub",null,[s("mi",null,"ν"),s("mn",null,"0")])]),s("annotation",{encoding:"application/x-tex"},"\\left.\\nu_0\\right): E_n=\\left(n+\\frac{1}{2}\\right) h \\nu_0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0576em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"n")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 多数の固有振動数の場合, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"∑"),s("mi",null,"h"),s("msub",null,[s("mi",null,"n"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"ν"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"E=\\sum h n_i \\nu_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"h"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.06366em"}},"ν"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0637em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(".")])],-1),d=s("h3",{id:"_11-6-トンネル効果",tabindex:"-1"},[a("11.6: トンネル効果 "),s("a",{class:"header-anchor",href:"#_11-6-トンネル効果","aria-label":'Permalink to "11.6: トンネル効果"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("トンネル効果: 幅 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" の障壁 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ")])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",null,"τ"),s("mo",null,"≈"),s("mi",{mathvariant:"normal"},"ℏ"),s("mo",{stretchy:"false"},"("),s("mi",null,"τ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\Gamma \\tau \\approx \\hbar(\\tau=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Γ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"ℏ"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"l"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Γ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m")])]),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"l / \\sqrt{\\Gamma / m})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mord"},"/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"Γ/"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.js b/assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.js rename to assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.js index b7c444e8..6508a94f 100644 --- a/assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.js +++ b/assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.js @@ -1 +1 @@ -import{_ as n,c as e,z as s,a,E as m,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 12","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","filePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/12.md"},c=l('

Formulas for IPhO 日本語版: Section 12 ​

12: Kepler の法則 ​

12.1: F & U ​

',3),h=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"F=G M m / r^2, U=-G M m / r \\text {. }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])],-1),o=l('

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

',5),g=s("ol",{start:"4"},[s("li",null,[a("Kepler の第三法則 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},": r^{-2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(" に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mn",null,"3"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{3}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(" 乗 に比例する :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"T"),s("mn",null,"1"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"T"),s("mn",null,"2"),s("mn",null,"2")]),s("mo",null,"="),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"1"),s("mn",null,"3")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"2"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"T_1^2 / T_2^2=a_1^3 / a_2^3 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])])])])])])])])],-1),u=s("h3",{id:"_12-5-楕円軌道",tabindex:"-1"},[a("12.5: 楕円軌道 "),s("a",{class:"header-anchor",href:"#_12-5-楕円軌道","aria-label":'Permalink to "12.5: 楕円軌道"'},"​")],-1),d=s("ol",{start:"5"},[s("li",null,[a("重力場中で楕円軌道を描く質点の全エネルギー "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"K"),s("mo",null,"+"),s("mi",null,"U"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(K+U)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mclose"},")")])])]),a(" :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"E=-G M m / 2 a ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a")])])])])])])],-1),y=s("h3",{id:"_12-6-楕円率",tabindex:"-1"},[a("12.6: 楕円率 "),s("a",{class:"header-anchor",href:"#_12-6-楕円率","aria-label":'Permalink to "12.6: 楕円率"'},"​")],-1),v=s("ol",null,[s("li",null,[a("楕円率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",null,"="),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon=d / a \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" の場合, 軌道は焦点をずらし た円形をしていると考えられる.")])],-1),b=s("h3",{id:"_12-7-楕円の性質",tabindex:"-1"},[a("12.7: 楕円の性質 "),s("a",{class:"header-anchor",href:"#_12-7-楕円の性質","aria-label":'Permalink to "12.7: 楕円の性質"'},"​")],-1),x=s("ol",{start:"2"},[s("li",null,[a("楕円の性質 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"l"),s("mn",null,"2")]),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,"（"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},": l_1+l_2=2 a （ l_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord cjk_fallback"},"（"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"l"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"l_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は焦点までの距 離). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\alpha_1=\\alpha_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" (一方の焦点から出た光は他方の焦点に 反射する). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"S"),s("mo",null,"="),s("mi",null,"π"),s("mi",null,"a"),s("mi",null,"b")]),s("annotation",{encoding:"application/x-tex"},"S=\\pi a b")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"πab")])])]),a(".")])],-1),_=s("h3",{id:"_12-8-円の中心",tabindex:"-1"},[a("12.8: 円の中心 "),s("a",{class:"header-anchor",href:"#_12-8-円の中心","aria-label":'Permalink to "12.8: 円の中心"'},"​")],-1),k=s("ol",{start:"3"},[s("li",null,"円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.")],-1),w={id:"_12-9-nge-lenz-ベクトル",tabindex:"-1"},z=s("a",{class:"header-anchor",href:"#_12-9-nge-lenz-ベクトル","aria-label":'Permalink to "12.9: nge-Lenz ベクトル "'},"​",-1),M=s("ol",{start:"4"},[s("li",null,[a("Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"e")]),s("annotation",{encoding:"application/x-tex"},"e")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"e")])])]),a(" に一致する.]："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"M")]),s("mrow",null,[s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m")])]),s("mo",null,"−"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{e}=\\frac{\\boldsymbol{v} \\times \\boldsymbol{M}}{G M m}-\\boldsymbol{e}_r=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0491em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3631em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])])])],-1);function f(P,T,L,K,S,G){const t=p("Badge");return i(),e("div",null,[c,h,o,g,u,d,y,v,b,x,_,k,s("h3",w,[a("12.9: nge-Lenz ベクトル "),m(t,{type:"tip",text:"supplemental"}),a(),z]),M])}const F=n(r,[["render",f]]);export{U as __pageData,F as default}; +import{_ as n,c as e,z as s,a,E as m,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 12","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","filePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/12.md"},c=l('

Formulas for IPhO 日本語版: Section 12 ​

12: Kepler の法則 ​

12.1: F & U ​

',3),h=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"F=G M m / r^2, U=-G M m / r \\text {. }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])],-1),o=l('

12.2: Kepler の第一法則 ​

  1. Kepler の第一法則 (2 質点の重力相互作用)：それぞ れの軌道は，系の質量中心に焦点を持つ楕円，双曲線, 放物線になる. これは Runge-Lenz ベクトルから得ら れる (ポイント 9).

12.3: Kepler の第二法則 ​

  1. Kepler の第二法則（角運動量の保存）: 中心力が働く 場にある質点について，その位置ベクトルは単位時間 に一定の面積を描く.

12.4: Kepler の第三法則 ​

',5),g=s("ol",{start:"4"},[s("li",null,[a("Kepler の第三法則 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},": r^{-2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(" に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mn",null,"3"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{3}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(" 乗 に比例する :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"T"),s("mn",null,"1"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"T"),s("mn",null,"2"),s("mn",null,"2")]),s("mo",null,"="),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"1"),s("mn",null,"3")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"2"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"T_1^2 / T_2^2=a_1^3 / a_2^3 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])])])])])])])])],-1),u=s("h3",{id:"_12-5-楕円軌道",tabindex:"-1"},[a("12.5: 楕円軌道 "),s("a",{class:"header-anchor",href:"#_12-5-楕円軌道","aria-label":'Permalink to "12.5: 楕円軌道"'},"​")],-1),d=s("ol",{start:"5"},[s("li",null,[a("重力場中で楕円軌道を描く質点の全エネルギー "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"K"),s("mo",null,"+"),s("mi",null,"U"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(K+U)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mclose"},")")])])]),a(" :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"E=-G M m / 2 a ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a")])])])])])])],-1),y=s("h3",{id:"_12-6-楕円率",tabindex:"-1"},[a("12.6: 楕円率 "),s("a",{class:"header-anchor",href:"#_12-6-楕円率","aria-label":'Permalink to "12.6: 楕円率"'},"​")],-1),v=s("ol",null,[s("li",null,[a("楕円率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",null,"="),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon=d / a \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" の場合, 軌道は焦点をずらし た円形をしていると考えられる.")])],-1),b=s("h3",{id:"_12-7-楕円の性質",tabindex:"-1"},[a("12.7: 楕円の性質 "),s("a",{class:"header-anchor",href:"#_12-7-楕円の性質","aria-label":'Permalink to "12.7: 楕円の性質"'},"​")],-1),x=s("ol",{start:"2"},[s("li",null,[a("楕円の性質 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"l"),s("mn",null,"2")]),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,"（"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},": l_1+l_2=2 a （ l_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord cjk_fallback"},"（"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"l"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"l_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は焦点までの距 離). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\alpha_1=\\alpha_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" (一方の焦点から出た光は他方の焦点に 反射する). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"S"),s("mo",null,"="),s("mi",null,"π"),s("mi",null,"a"),s("mi",null,"b")]),s("annotation",{encoding:"application/x-tex"},"S=\\pi a b")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"πab")])])]),a(".")])],-1),_=s("h3",{id:"_12-8-円の中心",tabindex:"-1"},[a("12.8: 円の中心 "),s("a",{class:"header-anchor",href:"#_12-8-円の中心","aria-label":'Permalink to "12.8: 円の中心"'},"​")],-1),k=s("ol",{start:"3"},[s("li",null,"円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.")],-1),w={id:"_12-9-nge-lenz-ベクトル",tabindex:"-1"},z=s("a",{class:"header-anchor",href:"#_12-9-nge-lenz-ベクトル","aria-label":'Permalink to "12.9: nge-Lenz ベクトル "'},"​",-1),M=s("ol",{start:"4"},[s("li",null,[a("Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"e")]),s("annotation",{encoding:"application/x-tex"},"e")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"e")])])]),a(" に一致する.]："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"M")]),s("mrow",null,[s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m")])]),s("mo",null,"−"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{e}=\\frac{\\boldsymbol{v} \\times \\boldsymbol{M}}{G M m}-\\boldsymbol{e}_r=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0491em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3631em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])])])],-1);function f(P,T,L,K,S,G){const t=p("Badge");return i(),e("div",null,[c,h,o,g,u,d,y,v,b,x,_,k,s("h3",w,[a("12.9: nge-Lenz ベクトル "),m(t,{type:"tip",text:"supplemental"}),a(),z]),M])}const F=n(r,[["render",f]]);export{U as __pageData,F as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.lean.js b/assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.lean.js rename to assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.lean.js index 2e6d23fc..d0a7814f 100644 --- a/assets/academic_physics_ipho-formulas-jpn_12.md.6b0b8711.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_12.md.bff53a1c.lean.js @@ -1 +1 @@ -import{_ as n,c as e,z as s,a,E as m,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 12","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","filePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/12.md"},c=l("",3),h=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"F=G M m / r^2, U=-G M m / r \\text {. }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])],-1),o=l("",5),g=s("ol",{start:"4"},[s("li",null,[a("Kepler の第三法則 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},": r^{-2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(" に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mn",null,"3"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{3}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(" 乗 に比例する :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"T"),s("mn",null,"1"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"T"),s("mn",null,"2"),s("mn",null,"2")]),s("mo",null,"="),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"1"),s("mn",null,"3")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"2"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"T_1^2 / T_2^2=a_1^3 / a_2^3 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])])])])])])])])],-1),u=s("h3",{id:"_12-5-楕円軌道",tabindex:"-1"},[a("12.5: 楕円軌道 "),s("a",{class:"header-anchor",href:"#_12-5-楕円軌道","aria-label":'Permalink to "12.5: 楕円軌道"'},"​")],-1),d=s("ol",{start:"5"},[s("li",null,[a("重力場中で楕円軌道を描く質点の全エネルギー "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"K"),s("mo",null,"+"),s("mi",null,"U"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(K+U)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mclose"},")")])])]),a(" :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"E=-G M m / 2 a ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a")])])])])])])],-1),y=s("h3",{id:"_12-6-楕円率",tabindex:"-1"},[a("12.6: 楕円率 "),s("a",{class:"header-anchor",href:"#_12-6-楕円率","aria-label":'Permalink to "12.6: 楕円率"'},"​")],-1),v=s("ol",null,[s("li",null,[a("楕円率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",null,"="),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon=d / a \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" の場合, 軌道は焦点をずらし た円形をしていると考えられる.")])],-1),b=s("h3",{id:"_12-7-楕円の性質",tabindex:"-1"},[a("12.7: 楕円の性質 "),s("a",{class:"header-anchor",href:"#_12-7-楕円の性質","aria-label":'Permalink to "12.7: 楕円の性質"'},"​")],-1),x=s("ol",{start:"2"},[s("li",null,[a("楕円の性質 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"l"),s("mn",null,"2")]),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,"（"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},": l_1+l_2=2 a （ l_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord cjk_fallback"},"（"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"l"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"l_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は焦点までの距 離). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\alpha_1=\\alpha_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" (一方の焦点から出た光は他方の焦点に 反射する). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"S"),s("mo",null,"="),s("mi",null,"π"),s("mi",null,"a"),s("mi",null,"b")]),s("annotation",{encoding:"application/x-tex"},"S=\\pi a b")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"πab")])])]),a(".")])],-1),_=s("h3",{id:"_12-8-円の中心",tabindex:"-1"},[a("12.8: 円の中心 "),s("a",{class:"header-anchor",href:"#_12-8-円の中心","aria-label":'Permalink to "12.8: 円の中心"'},"​")],-1),k=s("ol",{start:"3"},[s("li",null,"円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.")],-1),w={id:"_12-9-nge-lenz-ベクトル",tabindex:"-1"},z=s("a",{class:"header-anchor",href:"#_12-9-nge-lenz-ベクトル","aria-label":'Permalink to "12.9: nge-Lenz ベクトル "'},"​",-1),M=s("ol",{start:"4"},[s("li",null,[a("Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"e")]),s("annotation",{encoding:"application/x-tex"},"e")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"e")])])]),a(" に一致する.]："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"M")]),s("mrow",null,[s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m")])]),s("mo",null,"−"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{e}=\\frac{\\boldsymbol{v} \\times \\boldsymbol{M}}{G M m}-\\boldsymbol{e}_r=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0491em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3631em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])])])],-1);function f(P,T,L,K,S,G){const t=p("Badge");return i(),e("div",null,[c,h,o,g,u,d,y,v,b,x,_,k,s("h3",w,[a("12.9: nge-Lenz ベクトル "),m(t,{type:"tip",text:"supplemental"}),a(),z]),M])}const F=n(r,[["render",f]]);export{U as __pageData,F as default}; +import{_ as n,c as e,z as s,a,E as m,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 12","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","filePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/12.md"},c=l("",3),h=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"F=G M m / r^2, U=-G M m / r \\text {. }")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])],-1),o=l("",5),g=s("ol",{start:"4"},[s("li",null,[a("Kepler の第三法則 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"2")])])]),s("annotation",{encoding:"application/x-tex"},": r^{-2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"2")])])])])])])])])])])]),a(" に比例する力が働く場で楕円 軌道を描く複数の質点について, 周期は長半径の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mn",null,"3"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{3}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1901em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"3")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(" 乗 に比例する :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"T"),s("mn",null,"1"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"T"),s("mn",null,"2"),s("mn",null,"2")]),s("mo",null,"="),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"1"),s("mn",null,"3")]),s("mi",{mathvariant:"normal"},"/"),s("msubsup",null,[s("mi",null,"a"),s("mn",null,"2"),s("mn",null,"3")])]),s("annotation",{encoding:"application/x-tex"},"T_1^2 / T_2^2=a_1^3 / a_2^3 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"T"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])])])])])])])])],-1),u=s("h3",{id:"_12-5-楕円軌道",tabindex:"-1"},[a("12.5: 楕円軌道 "),s("a",{class:"header-anchor",href:"#_12-5-楕円軌道","aria-label":'Permalink to "12.5: 楕円軌道"'},"​")],-1),d=s("ol",{start:"5"},[s("li",null,[a("重力場中で楕円軌道を描く質点の全エネルギー "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"K"),s("mo",null,"+"),s("mi",null,"U"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(K+U)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mclose"},")")])])]),a(" :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mo",null,"−"),s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m"),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"E=-G M m / 2 a ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a")])])])])])])],-1),y=s("h3",{id:"_12-6-楕円率",tabindex:"-1"},[a("12.6: 楕円率 "),s("a",{class:"header-anchor",href:"#_12-6-楕円率","aria-label":'Permalink to "12.6: 楕円率"'},"​")],-1),v=s("ol",null,[s("li",null,[a("楕円率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",null,"="),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"≪"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon=d / a \\ll 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1")])])]),a(" の場合, 軌道は焦点をずらし た円形をしていると考えられる.")])],-1),b=s("h3",{id:"_12-7-楕円の性質",tabindex:"-1"},[a("12.7: 楕円の性質 "),s("a",{class:"header-anchor",href:"#_12-7-楕円の性質","aria-label":'Permalink to "12.7: 楕円の性質"'},"​")],-1),x=s("ol",{start:"2"},[s("li",null,[a("楕円の性質 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"l"),s("mn",null,"2")]),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,"（"),s("msub",null,[s("mi",null,"l"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},": l_1+l_2=2 a （ l_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord cjk_fallback"},"（"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"l"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"l_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8444em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0197em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は焦点までの距 離). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\alpha_1=\\alpha_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" (一方の焦点から出た光は他方の焦点に 反射する). "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"S"),s("mo",null,"="),s("mi",null,"π"),s("mi",null,"a"),s("mi",null,"b")]),s("annotation",{encoding:"application/x-tex"},"S=\\pi a b")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"πab")])])]),a(".")])],-1),_=s("h3",{id:"_12-8-円の中心",tabindex:"-1"},[a("12.8: 円の中心 "),s("a",{class:"header-anchor",href:"#_12-8-円の中心","aria-label":'Permalink to "12.8: 円の中心"'},"​")],-1),k=s("ol",{start:"3"},[s("li",null,"円とその円の中心に焦点をもつ楕円は長軸の部分での み接する.")],-1),w={id:"_12-9-nge-lenz-ベクトル",tabindex:"-1"},z=s("a",{class:"header-anchor",href:"#_12-9-nge-lenz-ベクトル","aria-label":'Permalink to "12.9: nge-Lenz ベクトル "'},"​",-1),M=s("ol",{start:"4"},[s("li",null,[a("Runge-Lenz ベクトル（楕円率ベクトル）[訳者注：こ のベクトルはむしろ離心率と関係する.そこでここで は離心率ベクトルとして知られるベクトルを代わりに 記す.これは焦点から近日点に向かう向きで，大きさ が離心率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"e")]),s("annotation",{encoding:"application/x-tex"},"e")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"e")])])]),a(" に一致する.]："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"M")]),s("mrow",null,[s("mi",null,"G"),s("mi",null,"M"),s("mi",null,"m")])]),s("mo",null,"−"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{e}=\\frac{\\boldsymbol{v} \\times \\boldsymbol{M}}{G M m}-\\boldsymbol{e}_r=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0491em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3631em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])])])],-1);function f(P,T,L,K,S,G){const t=p("Badge");return i(),e("div",null,[c,h,o,g,u,d,y,v,b,x,_,k,s("h3",w,[a("12.9: nge-Lenz ベクトル "),m(t,{type:"tip",text:"supplemental"}),a(),z]),M])}const F=n(r,[["render",f]]);export{U as __pageData,F as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.js b/assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.js rename to assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.js index 79696f2f..61e71351 100644 --- a/assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.js +++ b/assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.js @@ -1,4 +1,4 @@ -import{_ as t,c as m,z as s,a,E as n,R as e,o as i,G as p}from"./chunks/framework.a1bac067.js";const j=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 13","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/13.md","filePath":"academic/physics/ipho-formulas-jpn/13.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/13.md"},c=e('

Formulas for IPhO 日本語版: Section 13 ​

13: 相対性理論 ​

13.1:Lorentz 変換 ​

',3),h=s("ol",null,[s("li",null,[s("p",null,[a("Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"V"),s("mo",null,"="),s("mi",null,"V"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("mi",null,"β"),s("mo",null,"="),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{separator:"true"},","),s("mi",null,"γ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left.\\boldsymbol{V}=V \\boldsymbol{e}_x\\right): \\beta=V / c, \\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.25555em"}},"V")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"β"),s("mn",null,"2")])])]),s("mtext",null," として, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mspace",{width:"2em"}),s("mi",null,"c"),s("msup",null,[s("mi",null,"t"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"c"),s("mi",null,"t"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"c"),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"y")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"E"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"−"),s("mi",null,"β"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," ここで, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",null,"="),s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"m"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",{separator:"true"},","),s("mtext",null," etc. ")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & 1 / \\sqrt{1-\\beta^2} \\text { として, } \\\\ & \\qquad c t^{\\prime}=\\gamma(c t-\\beta x), x^{\\prime}=\\gamma(x-\\beta c t), y^{\\prime}=y \\\\ & E^{\\prime} / c=\\gamma\\left(E / c-\\beta p_x\\right), p_x^{\\prime}=\\gamma\\left(p_x-\\beta E / c\\right), p_y^{\\prime}=p_y \\\\ & \\text { ここで, } \\\\ & E=\\frac{m c^2}{\\sqrt{1-v^2 / c^2}}=m c^2+\\frac{1}{2} m v^2+\\cdots \\\\ & p_x=\\frac{m v_x}{\\sqrt{1-v^2 / c^2}}, \\text { etc. } \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.6485em","vertical-align":"-5.5742em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.0586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.0355em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.8844em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.6531em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.5742em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0067em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-2.9667em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as m,z as s,a,E as n,R as e,o as i,G as p}from"./chunks/framework.a1bac067.js";const j=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 13","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/13.md","filePath":"academic/physics/ipho-formulas-jpn/13.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/13.md"},c=e('

Formulas for IPhO 日本語版: Section 13 ​

13: 相対性理論 ​

13.1:Lorentz 変換 ​

',3),h=s("ol",null,[s("li",null,[s("p",null,[a("Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"V"),s("mo",null,"="),s("mi",null,"V"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("mi",null,"β"),s("mo",null,"="),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{separator:"true"},","),s("mi",null,"γ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left.\\boldsymbol{V}=V \\boldsymbol{e}_x\\right): \\beta=V / c, \\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.25555em"}},"V")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"β"),s("mn",null,"2")])])]),s("mtext",null," として, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mspace",{width:"2em"}),s("mi",null,"c"),s("msup",null,[s("mi",null,"t"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"c"),s("mi",null,"t"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"c"),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"y")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"E"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"−"),s("mi",null,"β"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," ここで, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",null,"="),s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"m"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",{separator:"true"},","),s("mtext",null," etc. ")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & 1 / \\sqrt{1-\\beta^2} \\text { として, } \\\\ & \\qquad c t^{\\prime}=\\gamma(c t-\\beta x), x^{\\prime}=\\gamma(x-\\beta c t), y^{\\prime}=y \\\\ & E^{\\prime} / c=\\gamma\\left(E / c-\\beta p_x\\right), p_x^{\\prime}=\\gamma\\left(p_x-\\beta E / c\\right), p_y^{\\prime}=p_y \\\\ & \\text { ここで, } \\\\ & E=\\frac{m c^2}{\\sqrt{1-v^2 / c^2}}=m c^2+\\frac{1}{2} m v^2+\\cdots \\\\ & p_x=\\frac{m v_x}{\\sqrt{1-v^2 / c^2}}, \\text { etc. } \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.6485em","vertical-align":"-5.5742em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.0586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.0355em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.8844em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.6531em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.5742em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0067em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-2.9667em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.lean.js b/assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.lean.js rename to assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.lean.js index 59247fae..c3cc0a89 100644 --- a/assets/academic_physics_ipho-formulas-jpn_13.md.b330f9cc.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_13.md.40dc3c65.lean.js @@ -1,4 +1,4 @@ -import{_ as t,c as m,z as s,a,E as n,R as e,o as i,G as p}from"./chunks/framework.a1bac067.js";const j=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 13","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/13.md","filePath":"academic/physics/ipho-formulas-jpn/13.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/13.md"},c=e("",3),h=s("ol",null,[s("li",null,[s("p",null,[a("Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"V"),s("mo",null,"="),s("mi",null,"V"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("mi",null,"β"),s("mo",null,"="),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{separator:"true"},","),s("mi",null,"γ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left.\\boldsymbol{V}=V \\boldsymbol{e}_x\\right): \\beta=V / c, \\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.25555em"}},"V")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"β"),s("mn",null,"2")])])]),s("mtext",null," として, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mspace",{width:"2em"}),s("mi",null,"c"),s("msup",null,[s("mi",null,"t"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"c"),s("mi",null,"t"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"c"),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"y")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"E"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"−"),s("mi",null,"β"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," ここで, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",null,"="),s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"m"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",{separator:"true"},","),s("mtext",null," etc. ")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & 1 / \\sqrt{1-\\beta^2} \\text { として, } \\\\ & \\qquad c t^{\\prime}=\\gamma(c t-\\beta x), x^{\\prime}=\\gamma(x-\\beta c t), y^{\\prime}=y \\\\ & E^{\\prime} / c=\\gamma\\left(E / c-\\beta p_x\\right), p_x^{\\prime}=\\gamma\\left(p_x-\\beta E / c\\right), p_y^{\\prime}=p_y \\\\ & \\text { ここで, } \\\\ & E=\\frac{m c^2}{\\sqrt{1-v^2 / c^2}}=m c^2+\\frac{1}{2} m v^2+\\cdots \\\\ & p_x=\\frac{m v_x}{\\sqrt{1-v^2 / c^2}}, \\text { etc. } \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.6485em","vertical-align":"-5.5742em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.0586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.0355em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.8844em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.6531em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.5742em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0067em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-2.9667em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as m,z as s,a,E as n,R as e,o as i,G as p}from"./chunks/framework.a1bac067.js";const j=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 13","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/13.md","filePath":"academic/physics/ipho-formulas-jpn/13.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/13.md"},c=e("",3),h=s("ol",null,[s("li",null,[s("p",null,[a("Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"V"),s("mo",null,"="),s("mi",null,"V"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",null,":"),s("mi",null,"β"),s("mo",null,"="),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{separator:"true"},","),s("mi",null,"γ"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\left.\\boldsymbol{V}=V \\boldsymbol{e}_x\\right): \\beta=V / c, \\gamma=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.25555em"}},"V")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"β"),s("mn",null,"2")])])]),s("mtext",null," として, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mspace",{width:"2em"}),s("mi",null,"c"),s("msup",null,[s("mi",null,"t"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"c"),s("mi",null,"t"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"c"),s("mi",null,"t"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("msup",null,[s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"y")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"E"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",null,"−"),s("mi",null,"β"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"x"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"γ"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"−"),s("mi",null,"β"),s("mi",null,"E"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"c"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"p"),s("mi",null,"y"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mtext",null," ここで, ")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",null,"="),s("mi",null,"m"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")]),s("mo",null,"+"),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mi",null,"m"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"+"),s("mo",null,"⋯")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"m"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])]),s("msqrt",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"−"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"c"),s("mn",null,"2")])])])]),s("mo",{separator:"true"},","),s("mtext",null," etc. ")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & 1 / \\sqrt{1-\\beta^2} \\text { として, } \\\\ & \\qquad c t^{\\prime}=\\gamma(c t-\\beta x), x^{\\prime}=\\gamma(x-\\beta c t), y^{\\prime}=y \\\\ & E^{\\prime} / c=\\gamma\\left(E / c-\\beta p_x\\right), p_x^{\\prime}=\\gamma\\left(p_x-\\beta E / c\\right), p_y^{\\prime}=p_y \\\\ & \\text { ここで, } \\\\ & E=\\frac{m c^2}{\\sqrt{1-v^2 / c^2}}=m c^2+\\frac{1}{2} m v^2+\\cdots \\\\ & p_x=\\frac{m v_x}{\\sqrt{1-v^2 / c^2}}, \\text { etc. } \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"11.6485em","vertical-align":"-5.5742em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-7.0586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-5.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.0355em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.8844em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"0.6531em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"5.5742em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"6.0742em"}},[s("span",{style:{top:"-8.5586em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.0067em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-2.9667em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.js b/assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.js similarity index 97% rename from assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.js rename to assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.js index c6e80409..a28242e6 100644 --- a/assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.js +++ b/assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.js @@ -1 +1 @@ -import{_ as e,c as t,o,z as a,a as s}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/2.md","filePath":"academic/physics/ipho-formulas-jpn/2.md","lastUpdated":1684858945000}'),l={name:"academic/physics/ipho-formulas-jpn/2.md"},n=a("h1",{id:"formulas-for-ipho-日本語版-section-2",tabindex:"-1"},[s("Formulas for IPhO 日本語版: Section 2 "),a("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-2","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 2"'},"​")],-1),i=a("h2",{id:"_2-一般的な推奨事",tabindex:"-1"},[s("2: 一般的な推奨事 "),a("a",{class:"header-anchor",href:"#_2-一般的な推奨事","aria-label":'Permalink to "2: 一般的な推奨事"'},"​")],-1),r=a("ol",null,[a("li",null,[s("全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"∞")]),a("annotation",{encoding:"application/x-tex"},"\\infty")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.4306em"}}),a("span",{class:"mord"},"∞")])])]),s(" ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.")]),a("li",null,"問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む."),a("li",null,"長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする."),a("li",null,"絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける."),a("li",null,"実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.")],-1),c=[n,i,r];function h(m,p,d,u,_,f){return o(),t("div",null,c)}const k=e(l,[["render",h]]);export{b as __pageData,k as default}; +import{_ as e,c as t,o,z as a,a as s}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/2.md","filePath":"academic/physics/ipho-formulas-jpn/2.md","lastUpdated":1684859551000}'),l={name:"academic/physics/ipho-formulas-jpn/2.md"},n=a("h1",{id:"formulas-for-ipho-日本語版-section-2",tabindex:"-1"},[s("Formulas for IPhO 日本語版: Section 2 "),a("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-2","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 2"'},"​")],-1),i=a("h2",{id:"_2-一般的な推奨事",tabindex:"-1"},[s("2: 一般的な推奨事 "),a("a",{class:"header-anchor",href:"#_2-一般的な推奨事","aria-label":'Permalink to "2: 一般的な推奨事"'},"​")],-1),r=a("ol",null,[a("li",null,[s("全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"∞")]),a("annotation",{encoding:"application/x-tex"},"\\infty")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.4306em"}}),a("span",{class:"mord"},"∞")])])]),s(" ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.")]),a("li",null,"問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む."),a("li",null,"長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする."),a("li",null,"絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける."),a("li",null,"実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.")],-1),c=[n,i,r];function h(m,p,d,u,_,f){return o(),t("div",null,c)}const k=e(l,[["render",h]]);export{b as __pageData,k as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.lean.js b/assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.lean.js similarity index 97% rename from assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.lean.js rename to assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.lean.js index c6e80409..a28242e6 100644 --- a/assets/academic_physics_ipho-formulas-jpn_2.md.3b1f7ddd.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_2.md.2b6166b8.lean.js @@ -1 +1 @@ -import{_ as e,c as t,o,z as a,a as s}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/2.md","filePath":"academic/physics/ipho-formulas-jpn/2.md","lastUpdated":1684858945000}'),l={name:"academic/physics/ipho-formulas-jpn/2.md"},n=a("h1",{id:"formulas-for-ipho-日本語版-section-2",tabindex:"-1"},[s("Formulas for IPhO 日本語版: Section 2 "),a("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-2","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 2"'},"​")],-1),i=a("h2",{id:"_2-一般的な推奨事",tabindex:"-1"},[s("2: 一般的な推奨事 "),a("a",{class:"header-anchor",href:"#_2-一般的な推奨事","aria-label":'Permalink to "2: 一般的な推奨事"'},"​")],-1),r=a("ol",null,[a("li",null,[s("全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"∞")]),a("annotation",{encoding:"application/x-tex"},"\\infty")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.4306em"}}),a("span",{class:"mord"},"∞")])])]),s(" ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.")]),a("li",null,"問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む."),a("li",null,"長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする."),a("li",null,"絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける."),a("li",null,"実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.")],-1),c=[n,i,r];function h(m,p,d,u,_,f){return o(),t("div",null,c)}const k=e(l,[["render",h]]);export{b as __pageData,k as default}; +import{_ as e,c as t,o,z as a,a as s}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 2","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/2.md","filePath":"academic/physics/ipho-formulas-jpn/2.md","lastUpdated":1684859551000}'),l={name:"academic/physics/ipho-formulas-jpn/2.md"},n=a("h1",{id:"formulas-for-ipho-日本語版-section-2",tabindex:"-1"},[s("Formulas for IPhO 日本語版: Section 2 "),a("a",{class:"header-anchor",href:"#formulas-for-ipho-日本語版-section-2","aria-label":'Permalink to "Formulas for IPhO 日本語版: Section 2"'},"​")],-1),i=a("h2",{id:"_2-一般的な推奨事",tabindex:"-1"},[s("2: 一般的な推奨事 "),a("a",{class:"header-anchor",href:"#_2-一般的な推奨事","aria-label":'Permalink to "2: 一般的な推奨事"'},"​")],-1),r=a("ol",null,[a("li",null,[s("全ての計算式の正しさを確かめる：a) 次元を調べる. b) 簡単で特別な場合を調べる（2 つの変数が等しい, 1 つの変数が 0 または "),a("span",{class:"katex"},[a("span",{class:"katex-mathml"},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("semantics",null,[a("mrow",null,[a("mi",{mathvariant:"normal"},"∞")]),a("annotation",{encoding:"application/x-tex"},"\\infty")])])]),a("span",{class:"katex-html","aria-hidden":"true"},[a("span",{class:"base"},[a("span",{class:"strut",style:{height:"0.4306em"}}),a("span",{class:"mord"},"∞")])])]),s(" ). c) 解の定性的な挙動の妥当 性を調ベる. 2.もし問題文中に驚くベき偶然の一致があれば（例えば 2 つのものが同じ), 解答の鍵はそこにあるかもしれ ない.")]),a("li",null,"問題文中の推奨事項をよく読む. 些細な部分に重要な 情報が含まれている場合があるので，問題文の文言に 注意する. かなり時間をかけても問題が解けない場合 は, 問題を誤解しているかもしれないので, もう一度 問題文を読む."),a("li",null,"長くて時間のかかる計算は, 簡略化しなければならな い始めの方程式を全て書き出したのち, 最後（他の全 てが終わったとき) まで先送りする."),a("li",null,"絶望的に難しいと思われる問題でも，たいてい非常に シンプルな解法がある.オリンピックの問題に限って 言えば，絶対に解ける."),a("li",null,"実験では, a) 測定するほどの時問が無いとしても, 実 験計画の概略を書く，b) 結果の正確さを高める方法を 考える，c) 測定した值を全て（表として）書き出す.")],-1),c=[n,i,r];function h(m,p,d,u,_,f){return o(),t("div",null,c)}const k=e(l,[["render",h]]);export{b as __pageData,k as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.js b/assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.js rename to assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.js index 78898d26..3abba9a1 100644 --- a/assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.js +++ b/assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.js @@ -1 +1 @@ -import{_ as n,c as m,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/3.md","filePath":"academic/physics/ipho-formulas-jpn/3.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/3.md"},c=l('

Formulas for IPhO 日本語版: Section 3 ​

3: 運動学 ​

3.1: 質点 ​

',3),o=s("ol",null,[s("li",null,[a("質点または剛体の並進運動の場合（積分 → グラフの下 の面積）："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"x"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"v"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mtext",null," など "),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"d"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")])])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"a"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"t"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"a"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("mfrac",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{v}=\\frac{\\mathrm{d} \\boldsymbol{x}}{\\mathrm{d} t}, \\boldsymbol{x}=\\int \\boldsymbol{v} \\mathrm{d} t\\left(x=\\int v_x \\mathrm{~d} t \\text { など }\\right) \\\\ \\boldsymbol{a}=\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{d} t}=\\frac{\\mathrm{d}^2 \\boldsymbol{x}}{\\mathrm{d} t^2}, \\boldsymbol{v}=\\int \\boldsymbol{a} \\mathrm{d} t \\\\ t=\\int v_x^{-1} \\mathrm{~d} x=\\int a_x^{-1} \\mathrm{~d} v_x, x=\\int \\frac{v_x}{a_x} \\mathrm{~d} v_x \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8756em","vertical-align":"-3.6878em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1878em"}},[s("span",{style:{top:"-6.2289em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"など"),s("span",{class:"mord"}," ")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])])])]),s("span",{style:{top:"-3.4878em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4911em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-0.9655em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.836em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6878em"}},[s("span")])])])])])])])])])])]),a(" もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"0")]),s("mi",null,"t"),s("mo",null,"+"),s("mi",null,"a"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"−"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"x=v_0 t+a t^2 / 2=\\left(v^2-v_0^2\\right) / 2 a \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7651em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])])])],-1),h=s("h3",{id:"_3-2-回転運動",tabindex:"-1"},[a("3.2: 回転運動 "),s("a",{class:"header-anchor",href:"#_3-2-回転運動","aria-label":'Permalink to "3.2: 回転運動"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("回転運動は, 並進運動と似ていて："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"ω")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("mi",null,"ε"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",{mathvariant:"bold-italic"},"a")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"τ"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"v"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"n"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\omega & =\\mathrm{d} \\varphi / \\mathrm{d} t, \\varepsilon=\\mathrm{d} \\omega / \\mathrm{d} t \\\\ \\boldsymbol{a} & =\\boldsymbol{\\tau} \\mathrm{d} v / \\mathrm{d} t+\\boldsymbol{n} v^2 / R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.13472em"}},"τ")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"n")])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),d=l('

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

',3),u=s("ol",{start:"4"},[s("li",null,[a("剛体の運動: "),s("ul",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"A")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mi",null,"B")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"v_A \\cos \\alpha=v_B \\cos \\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" ここで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は剛体上の点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の速度, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"α")]),s("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"\\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" が直線 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" となす角.")]),s("li",null,[a("瞬間回転中心 (#質点の軌道 の曲率中心）は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"b")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{b}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])])]),a(" に下ろした垂線の交点. 又は もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")]),s("mo",null,"⊥"),s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A, \\boldsymbol{v}_B \\perp A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" ならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の先端を結ぶ 直線と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の交点.")])])])],-1),y=s("h3",{id:"_3-5-非慣性系",tabindex:"-1"},[a("3.5: 非慣性系 "),s("a",{class:"header-anchor",href:"#_3-5-非慣性系","aria-label":'Permalink to "3.5: 非慣性系"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("非慣性系："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.16em",columnalign:"right",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mspace",{width:"1em"}),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"1")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"R"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mtext",null," ここで, "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"⊥"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"."),s("mtext",null," もし "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",null,"="),s("mn",null,"0"),s("mtext",null," なら "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"="),s("mn",null,"0.")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{array}{r} \\quad \\boldsymbol{v}_2=\\boldsymbol{v}_0+\\boldsymbol{v}_1, \\boldsymbol{a}_2=\\boldsymbol{a}_0+\\boldsymbol{a}_1+\\omega^2 \\boldsymbol{R}+\\boldsymbol{a}_{C o r} \\\\ \\text { ここで, } \\boldsymbol{a}_{C o r} \\perp \\boldsymbol{v}_1 . \\text { もし } \\boldsymbol{v}_1=0 \\text { なら } \\boldsymbol{a}_{C o r}=0 . \\end{array} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"arraycolsep",style:{width:"0.5em"}}),s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.45em"}},[s("span",{style:{top:"-3.61em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"ここで"),s("span",{class:"mord"},", ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"."),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"もし"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"なら"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.95em"}},[s("span")])])])]),s("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])])])])],-1),b={id:"_3-6-弾道問題",tabindex:"-1"},x=s("a",{class:"header-anchor",href:"#_3-6-弾道問題","aria-label":'Permalink to "3.6: 弾道問題 "'},"​",-1),z=s("ol",{start:"6"},[s("li",null,[a("弾道問題：到達可能な範囲は"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"≤"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"2"),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"−"),s("mi",null,"g"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y \\leq v_0^2 /(2 g)-g x^2 /\\left(2 v_0^2\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8304em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≤"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])])])]),a(" 最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.")])],-1),w=l('

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
',4);function _(k,f,M,A,B,L){const t=p("Badge");return i(),m("div",null,[c,o,h,g,d,u,y,v,s("h3",b,[a("3.6: 弾道問題 "),e(t,{type:"tip",text:"supplemental"}),a(),x]),z,w])}const q=n(r,[["render",_]]);export{T as __pageData,q as default}; +import{_ as n,c as m,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/3.md","filePath":"academic/physics/ipho-formulas-jpn/3.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/3.md"},c=l('

Formulas for IPhO 日本語版: Section 3 ​

3: 運動学 ​

3.1: 質点 ​

',3),o=s("ol",null,[s("li",null,[a("質点または剛体の並進運動の場合（積分 → グラフの下 の面積）："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"x"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"v"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mtext",null," など "),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"d"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")])])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"a"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"t"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"a"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("mfrac",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{v}=\\frac{\\mathrm{d} \\boldsymbol{x}}{\\mathrm{d} t}, \\boldsymbol{x}=\\int \\boldsymbol{v} \\mathrm{d} t\\left(x=\\int v_x \\mathrm{~d} t \\text { など }\\right) \\\\ \\boldsymbol{a}=\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{d} t}=\\frac{\\mathrm{d}^2 \\boldsymbol{x}}{\\mathrm{d} t^2}, \\boldsymbol{v}=\\int \\boldsymbol{a} \\mathrm{d} t \\\\ t=\\int v_x^{-1} \\mathrm{~d} x=\\int a_x^{-1} \\mathrm{~d} v_x, x=\\int \\frac{v_x}{a_x} \\mathrm{~d} v_x \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8756em","vertical-align":"-3.6878em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1878em"}},[s("span",{style:{top:"-6.2289em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"など"),s("span",{class:"mord"}," ")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])])])]),s("span",{style:{top:"-3.4878em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4911em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-0.9655em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.836em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6878em"}},[s("span")])])])])])])])])])])]),a(" もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"0")]),s("mi",null,"t"),s("mo",null,"+"),s("mi",null,"a"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"−"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"x=v_0 t+a t^2 / 2=\\left(v^2-v_0^2\\right) / 2 a \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7651em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])])])],-1),h=s("h3",{id:"_3-2-回転運動",tabindex:"-1"},[a("3.2: 回転運動 "),s("a",{class:"header-anchor",href:"#_3-2-回転運動","aria-label":'Permalink to "3.2: 回転運動"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("回転運動は, 並進運動と似ていて："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"ω")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("mi",null,"ε"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",{mathvariant:"bold-italic"},"a")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"τ"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"v"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"n"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\omega & =\\mathrm{d} \\varphi / \\mathrm{d} t, \\varepsilon=\\mathrm{d} \\omega / \\mathrm{d} t \\\\ \\boldsymbol{a} & =\\boldsymbol{\\tau} \\mathrm{d} v / \\mathrm{d} t+\\boldsymbol{n} v^2 / R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.13472em"}},"τ")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"n")])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),d=l('

3.3: 曲線運動 ​

  1. 曲線運動は，ポイント 1 と同じだが，ベクトルは線速 度，加速度，経路長に置き換える.

3.4: 剛体の運動 ​

',3),u=s("ol",{start:"4"},[s("li",null,[a("剛体の運動: "),s("ul",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"A")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mi",null,"B")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"v_A \\cos \\alpha=v_B \\cos \\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" ここで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は剛体上の点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の速度, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"α")]),s("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"\\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" が直線 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" となす角.")]),s("li",null,[a("瞬間回転中心 (#質点の軌道 の曲率中心）は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"b")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{b}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])])]),a(" に下ろした垂線の交点. 又は もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")]),s("mo",null,"⊥"),s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A, \\boldsymbol{v}_B \\perp A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" ならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の先端を結ぶ 直線と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の交点.")])])])],-1),y=s("h3",{id:"_3-5-非慣性系",tabindex:"-1"},[a("3.5: 非慣性系 "),s("a",{class:"header-anchor",href:"#_3-5-非慣性系","aria-label":'Permalink to "3.5: 非慣性系"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("非慣性系："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.16em",columnalign:"right",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mspace",{width:"1em"}),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"1")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"R"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mtext",null," ここで, "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"⊥"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"."),s("mtext",null," もし "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",null,"="),s("mn",null,"0"),s("mtext",null," なら "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"="),s("mn",null,"0.")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{array}{r} \\quad \\boldsymbol{v}_2=\\boldsymbol{v}_0+\\boldsymbol{v}_1, \\boldsymbol{a}_2=\\boldsymbol{a}_0+\\boldsymbol{a}_1+\\omega^2 \\boldsymbol{R}+\\boldsymbol{a}_{C o r} \\\\ \\text { ここで, } \\boldsymbol{a}_{C o r} \\perp \\boldsymbol{v}_1 . \\text { もし } \\boldsymbol{v}_1=0 \\text { なら } \\boldsymbol{a}_{C o r}=0 . \\end{array} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"arraycolsep",style:{width:"0.5em"}}),s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.45em"}},[s("span",{style:{top:"-3.61em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"ここで"),s("span",{class:"mord"},", ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"."),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"もし"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"なら"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.95em"}},[s("span")])])])]),s("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])])])])],-1),b={id:"_3-6-弾道問題",tabindex:"-1"},x=s("a",{class:"header-anchor",href:"#_3-6-弾道問題","aria-label":'Permalink to "3.6: 弾道問題 "'},"​",-1),z=s("ol",{start:"6"},[s("li",null,[a("弾道問題：到達可能な範囲は"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"≤"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"2"),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"−"),s("mi",null,"g"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y \\leq v_0^2 /(2 g)-g x^2 /\\left(2 v_0^2\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8304em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≤"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])])])]),a(" 最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.")])],-1),w=l('

3.7: 最短経路 ​

  1. 最短経路を求めるには，Fermat と Huygens の原理が 使える.

3.8: ベクトル ​

  1. ベクトル（速度，加速度）を求めるには，その向きと （場合によっては傾いた）ある軸への射影を求めれば 充分.
',4);function _(k,f,M,A,B,L){const t=p("Badge");return i(),m("div",null,[c,o,h,g,d,u,y,v,s("h3",b,[a("3.6: 弾道問題 "),e(t,{type:"tip",text:"supplemental"}),a(),x]),z,w])}const q=n(r,[["render",_]]);export{T as __pageData,q as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.lean.js b/assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.lean.js rename to assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.lean.js index 2166f64a..961e0221 100644 --- a/assets/academic_physics_ipho-formulas-jpn_3.md.417386cd.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_3.md.24fb7253.lean.js @@ -1 +1 @@ -import{_ as n,c as m,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/3.md","filePath":"academic/physics/ipho-formulas-jpn/3.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/3.md"},c=l("",3),o=s("ol",null,[s("li",null,[a("質点または剛体の並進運動の場合（積分 → グラフの下 の面積）："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"x"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"v"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mtext",null," など "),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"d"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")])])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"a"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"t"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"a"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("mfrac",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{v}=\\frac{\\mathrm{d} \\boldsymbol{x}}{\\mathrm{d} t}, \\boldsymbol{x}=\\int \\boldsymbol{v} \\mathrm{d} t\\left(x=\\int v_x \\mathrm{~d} t \\text { など }\\right) \\\\ \\boldsymbol{a}=\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{d} t}=\\frac{\\mathrm{d}^2 \\boldsymbol{x}}{\\mathrm{d} t^2}, \\boldsymbol{v}=\\int \\boldsymbol{a} \\mathrm{d} t \\\\ t=\\int v_x^{-1} \\mathrm{~d} x=\\int a_x^{-1} \\mathrm{~d} v_x, x=\\int \\frac{v_x}{a_x} \\mathrm{~d} v_x \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8756em","vertical-align":"-3.6878em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1878em"}},[s("span",{style:{top:"-6.2289em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"など"),s("span",{class:"mord"}," ")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])])])]),s("span",{style:{top:"-3.4878em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4911em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-0.9655em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.836em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6878em"}},[s("span")])])])])])])])])])])]),a(" もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"0")]),s("mi",null,"t"),s("mo",null,"+"),s("mi",null,"a"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"−"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"x=v_0 t+a t^2 / 2=\\left(v^2-v_0^2\\right) / 2 a \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7651em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])])])],-1),h=s("h3",{id:"_3-2-回転運動",tabindex:"-1"},[a("3.2: 回転運動 "),s("a",{class:"header-anchor",href:"#_3-2-回転運動","aria-label":'Permalink to "3.2: 回転運動"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("回転運動は, 並進運動と似ていて："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"ω")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("mi",null,"ε"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",{mathvariant:"bold-italic"},"a")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"τ"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"v"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"n"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\omega & =\\mathrm{d} \\varphi / \\mathrm{d} t, \\varepsilon=\\mathrm{d} \\omega / \\mathrm{d} t \\\\ \\boldsymbol{a} & =\\boldsymbol{\\tau} \\mathrm{d} v / \\mathrm{d} t+\\boldsymbol{n} v^2 / R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.13472em"}},"τ")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"n")])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),d=l("",3),u=s("ol",{start:"4"},[s("li",null,[a("剛体の運動: "),s("ul",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"A")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mi",null,"B")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"v_A \\cos \\alpha=v_B \\cos \\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" ここで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は剛体上の点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の速度, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"α")]),s("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"\\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" が直線 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" となす角.")]),s("li",null,[a("瞬間回転中心 (#質点の軌道 の曲率中心）は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"b")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{b}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])])]),a(" に下ろした垂線の交点. 又は もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")]),s("mo",null,"⊥"),s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A, \\boldsymbol{v}_B \\perp A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" ならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の先端を結ぶ 直線と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の交点.")])])])],-1),y=s("h3",{id:"_3-5-非慣性系",tabindex:"-1"},[a("3.5: 非慣性系 "),s("a",{class:"header-anchor",href:"#_3-5-非慣性系","aria-label":'Permalink to "3.5: 非慣性系"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("非慣性系："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.16em",columnalign:"right",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mspace",{width:"1em"}),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"1")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"R"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mtext",null," ここで, "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"⊥"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"."),s("mtext",null," もし "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",null,"="),s("mn",null,"0"),s("mtext",null," なら "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"="),s("mn",null,"0.")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{array}{r} \\quad \\boldsymbol{v}_2=\\boldsymbol{v}_0+\\boldsymbol{v}_1, \\boldsymbol{a}_2=\\boldsymbol{a}_0+\\boldsymbol{a}_1+\\omega^2 \\boldsymbol{R}+\\boldsymbol{a}_{C o r} \\\\ \\text { ここで, } \\boldsymbol{a}_{C o r} \\perp \\boldsymbol{v}_1 . \\text { もし } \\boldsymbol{v}_1=0 \\text { なら } \\boldsymbol{a}_{C o r}=0 . \\end{array} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"arraycolsep",style:{width:"0.5em"}}),s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.45em"}},[s("span",{style:{top:"-3.61em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"ここで"),s("span",{class:"mord"},", ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"."),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"もし"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"なら"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.95em"}},[s("span")])])])]),s("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])])])])],-1),b={id:"_3-6-弾道問題",tabindex:"-1"},x=s("a",{class:"header-anchor",href:"#_3-6-弾道問題","aria-label":'Permalink to "3.6: 弾道問題 "'},"​",-1),z=s("ol",{start:"6"},[s("li",null,[a("弾道問題：到達可能な範囲は"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"≤"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"2"),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"−"),s("mi",null,"g"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y \\leq v_0^2 /(2 g)-g x^2 /\\left(2 v_0^2\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8304em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≤"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])])])]),a(" 最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.")])],-1),w=l("",4);function _(k,f,M,A,B,L){const t=p("Badge");return i(),m("div",null,[c,o,h,g,d,u,y,v,s("h3",b,[a("3.6: 弾道問題 "),e(t,{type:"tip",text:"supplemental"}),a(),x]),z,w])}const q=n(r,[["render",_]]);export{T as __pageData,q as default}; +import{_ as n,c as m,z as s,a,E as e,R as l,o as i,G as p}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 3","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/3.md","filePath":"academic/physics/ipho-formulas-jpn/3.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/3.md"},c=l("",3),o=s("ol",null,[s("li",null,[a("質点または剛体の並進運動の場合（積分 → グラフの下 の面積）："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"x"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"v"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mtext",null," など "),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"v")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"d"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"x")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")])])]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"="),s("mo",null,"∫"),s("mi",{mathvariant:"bold-italic"},"a"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"t"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("msubsup",null,[s("mi",null,"a"),s("mi",null,"x"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("mi",null,"x"),s("mo",null,"="),s("mo",null,"∫"),s("mfrac",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"x")]),s("msub",null,[s("mi",null,"a"),s("mi",null,"x")])]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("msub",null,[s("mi",null,"v"),s("mi",null,"x")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{v}=\\frac{\\mathrm{d} \\boldsymbol{x}}{\\mathrm{d} t}, \\boldsymbol{x}=\\int \\boldsymbol{v} \\mathrm{d} t\\left(x=\\int v_x \\mathrm{~d} t \\text { など }\\right) \\\\ \\boldsymbol{a}=\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{d} t}=\\frac{\\mathrm{d}^2 \\boldsymbol{x}}{\\mathrm{d} t^2}, \\boldsymbol{v}=\\int \\boldsymbol{a} \\mathrm{d} t \\\\ t=\\int v_x^{-1} \\mathrm{~d} x=\\int a_x^{-1} \\mathrm{~d} v_x, x=\\int \\frac{v_x}{a_x} \\mathrm{~d} v_x \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8756em","vertical-align":"-3.6878em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1878em"}},[s("span",{style:{top:"-6.2289em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"など"),s("span",{class:"mord"}," ")]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])])])]),s("span",{style:{top:"-3.4878em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4911em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-0.9655em"}},[s("span",{class:"pstrut",style:{height:"3.4911em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.836em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6878em"}},[s("span")])])])])])])])])])])]),a(" もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" が定数ならば, これらの積分は簡単に求めるこ とができて, 例えば"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"0")]),s("mi",null,"t"),s("mo",null,"+"),s("mi",null,"a"),s("msup",null,[s("mi",null,"t"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mo",null,"−"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2"),s("mi",null,"a"),s("mtext",null,". ")]),s("annotation",{encoding:"application/x-tex"},"x=v_0 t+a t^2 / 2=\\left(v^2-v_0^2\\right) / 2 a \\text {. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7651em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"t"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/2"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord text"},[s("span",{class:"mord"},". ")])])])])])])])],-1),h=s("h3",{id:"_3-2-回転運動",tabindex:"-1"},[a("3.2: 回転運動 "),s("a",{class:"header-anchor",href:"#_3-2-回転運動","aria-label":'Permalink to "3.2: 回転運動"'},"​")],-1),g=s("ol",{start:"2"},[s("li",null,[a("回転運動は, 並進運動と似ていて："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"ω")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("mi",null,"ε"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ω"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",{mathvariant:"bold-italic"},"a")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"τ"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"v"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"n"),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} \\omega & =\\mathrm{d} \\varphi / \\mathrm{d} t, \\varepsilon=\\mathrm{d} \\omega / \\mathrm{d} t \\\\ \\boldsymbol{a} & =\\boldsymbol{\\tau} \\mathrm{d} v / \\mathrm{d} t+\\boldsymbol{n} v^2 / R \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.9221em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.13472em"}},"τ")])]),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"n")])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),d=l("",3),u=s("ol",{start:"4"},[s("li",null,[a("剛体の運動: "),s("ul",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"v"),s("mi",null,"A")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"α"),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mi",null,"B")]),s("mi",null,"cos"),s("mo",null,"⁡"),s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"v_A \\cos \\alpha=v_B \\cos \\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" ここで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は剛体上の点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の速度, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"α")]),s("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"β")]),s("annotation",{encoding:"application/x-tex"},"\\beta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),a(" は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" が直線 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" となす角.")]),s("li",null,[a("瞬間回転中心 (#質点の軌道 の曲率中心）は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"a")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"b")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{b}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"b")])])])])]),a(" に下ろした垂線の交点. 又は もし "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")]),s("mo",null,"⊥"),s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A, \\boldsymbol{v}_B \\perp A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" ならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"A")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"A")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"B")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05017em"}},"B")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の先端を結ぶ 直線と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"A B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" の交点.")])])])],-1),y=s("h3",{id:"_3-5-非慣性系",tabindex:"-1"},[a("3.5: 非慣性系 "),s("a",{class:"header-anchor",href:"#_3-5-非慣性系","aria-label":'Permalink to "3.5: 非慣性系"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("非慣性系："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.16em",columnalign:"right",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mspace",{width:"1em"}),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"0")]),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mn",null,"1")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mi",{mathvariant:"bold-italic"},"R"),s("mo",null,"+"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"false"},[s("mrow",null,[s("mtext",null," ここで, "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"⊥"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"."),s("mtext",null," もし "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mn",null,"1")]),s("mo",null,"="),s("mn",null,"0"),s("mtext",null," なら "),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"a"),s("mrow",null,[s("mi",null,"C"),s("mi",null,"o"),s("mi",null,"r")])]),s("mo",null,"="),s("mn",null,"0.")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{array}{r} \\quad \\boldsymbol{v}_2=\\boldsymbol{v}_0+\\boldsymbol{v}_1, \\boldsymbol{a}_2=\\boldsymbol{a}_0+\\boldsymbol{a}_1+\\omega^2 \\boldsymbol{R}+\\boldsymbol{a}_{C o r} \\\\ \\text { ここで, } \\boldsymbol{a}_{C o r} \\perp \\boldsymbol{v}_1 . \\text { もし } \\boldsymbol{v}_1=0 \\text { なら } \\boldsymbol{a}_{C o r}=0 . \\end{array} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"arraycolsep",style:{width:"0.5em"}}),s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.45em"}},[s("span",{style:{top:"-3.61em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-2.41em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"ここで"),s("span",{class:"mord"},", ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"⊥"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"."),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"もし"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mord text"},[s("span",{class:"mord"}," "),s("span",{class:"mord cjk_fallback"},"なら"),s("span",{class:"mord"}," ")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"or")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"0.")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.95em"}},[s("span")])])])]),s("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])])])])],-1),b={id:"_3-6-弾道問題",tabindex:"-1"},x=s("a",{class:"header-anchor",href:"#_3-6-弾道問題","aria-label":'Permalink to "3.6: 弾道問題 "'},"​",-1),z=s("ol",{start:"6"},[s("li",null,[a("弾道問題：到達可能な範囲は"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"≤"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mo",{stretchy:"false"},"("),s("mn",null,"2"),s("mi",null,"g"),s("mo",{stretchy:"false"},")"),s("mo",null,"−"),s("mi",null,"g"),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msubsup",null,[s("mi",null,"v"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"y \\leq v_0^2 /(2 g)-g x^2 /\\left(2 v_0^2\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8304em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≤"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1141em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2141em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])])])]),a(" 最適な弾道では, 初速度と終速（衝突時の速度）が垂直 になる.")])],-1),w=l("",4);function _(k,f,M,A,B,L){const t=p("Badge");return i(),m("div",null,[c,o,h,g,d,u,y,v,s("h3",b,[a("3.6: 弾道問題 "),e(t,{type:"tip",text:"supplemental"}),a(),x]),z,w])}const q=n(r,[["render",_]]);export{T as __pageData,q as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.js b/assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.js rename to assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.js index 3b449518..3edc2d30 100644 --- a/assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.js +++ b/assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.js @@ -1,4 +1,4 @@ -import{_ as m,c as e,z as s,a,E as t,R as n,o as i,G as p}from"./chunks/framework.a1bac067.js";const gs=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 4","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/4.md","filePath":"academic/physics/ipho-formulas-jpn/4.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/4.md"},c=n('

Formulas for IPhO 日本語版: Section 4 ​

4: 力学 ​

4.1: 剛体の二次元的な平衡 ​

  1. 剛体の二次元的な平衡 : 力についての 2 つの式とトル クについての 1 つの式. 1 （又は 2 ）個の力についての 式は 1（又は 2）個のトルクについての式で代用でき る. トルクの方が良い場合が多く，原点を適切に選択 することで「退屈な」力を消すことができる. もし 2 点 のみに力がかかっているならば，（正味の）力がかかっ ている直線は一致する. 3 点であれば, 3 つの直線は 1 点で交わる.

4.2: 垂直抗力 ​

',5),h=s("ol",{start:"2"},[s("li",null,[a("垂直抗力と摩擦力は 1 つの力に合成でき, 垂直抗力に 対して "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"arctan"),s("mo",null,"⁡"),s("mi",null,"μ")]),s("annotation",{encoding:"application/x-tex"},"\\arctan \\mu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},"arctan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"μ")])])]),a(" の角度で接触点に加わる.")])],-1),o=s("h3",{id:"_4-3-並進運動と回転運動",tabindex:"-1"},[a("4.3: 並進運動と回転運動 "),s("a",{class:"header-anchor",href:"#_4-3-並進運動と回転運動","aria-label":'Permalink to "4.3: 並進運動と回転運動"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("並進運動と回転運動についての Newton の第二法則："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"m"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"ε"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=m \\boldsymbol{a}, \\boldsymbol{M}=I \\boldsymbol{\\varepsilon} \\quad(\\boldsymbol{M}=\\boldsymbol{r} \\times \\boldsymbol{F}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8805em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"ε")])]),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mclose"},")")])])])])]),a(" 二次元の場合には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"M")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε")])])]),a(" は本質的にスカラーで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"l"),s("mo",null,"="),s("msub",null,[s("mi",null,"F"),s("mi",null,"t")]),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"M=F l=F_t r(l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"Fl"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2806em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"t")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" は力のうでの長さ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_4-4-一般化座標",tabindex:"-1"},[a("4.4: 一般化座標 "),s("a",{class:"header-anchor",href:"#_4-4-一般化座標","aria-label":'Permalink to "4.4: 一般化座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("一般化座標. 系の状態が 1 つの変数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"\\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(" とその時間微分 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"˙")])]),s("annotation",{encoding:"application/x-tex"},"\\dot{\\xi}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.0278em"}},[s("span",{class:"mord"},"˙")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(" で表され，ポテンシャルエネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"U=U(\\xi)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")")])])]),a(", 運動エネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"K"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"ξ"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")]),s("annotation",{encoding:"application/x-tex"},"K=\\mu \\xi^2 / 2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2")])])]),a(" であるならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"μ"),s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"¨")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\mu \\ddot{\\xi}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"¨")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"-\\mathrm{d} U(\\xi) / \\mathrm{d} \\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(". (したがって並進運動では, 力はポテン シャルエネルギーの微分)")])],-1),y=s("h3",{id:"_4-5-系質点",tabindex:"-1"},[a("4.5: 系質点 "),s("a",{class:"header-anchor",href:"#_4-5-系質点","aria-label":'Permalink to "4.5: 系質点"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("系が質点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"m"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"m_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" で構成されているとき："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"c")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mi",{mathvariant:"normal"},"/"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"i"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msubsup",null,[s("mi",null,"x"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"y"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"∫"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"y"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"m")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\boldsymbol{r}_c=\\sum m_i \\boldsymbol{r}_i / \\sum m_j, \\boldsymbol{P}=\\sum m_i \\boldsymbol{v}_i \\\\ & \\boldsymbol{L}=\\sum m_i \\boldsymbol{r}_i \\times \\boldsymbol{v}_i, K=\\sum m_i v_i^2 / 2 \\\\ & I_z=\\sum m_i\\left(x_i^2+y_i^2\\right)=\\int\\left(x^2+y^2\\right) \\mathrm{d} m \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8223em","vertical-align":"-3.6611em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_4-6-質量中心の速度",tabindex:"-1"},[a("4.6: 質量中心の速度 "),s("a",{class:"header-anchor",href:"#_4-6-質量中心の速度","aria-label":'Permalink to "4.6: 質量中心の速度"'},"​")],-1),x=s("ol",{start:"6"},[s("li",null,[a("質量中心の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" であるような系 (添え字 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" は質量 中心についての物理量であることを示す）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"R"),s("mi",null,"c")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("msub",null,[s("mi",null,"K"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"c"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},".")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{L}=\\boldsymbol{L}_c+M_{\\Sigma} \\boldsymbol{R}_c \\times \\boldsymbol{v}_c, K=K_c+M_{\\Sigma} v_c^2 / 2 \\\\ \\boldsymbol{P}=\\boldsymbol{P}_c+M_{\\Sigma} \\boldsymbol{v}_c . \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_4-7-steiner-定理",tabindex:"-1"},[a("4.7: Steiner 定理 "),s("a",{class:"header-anchor",href:"#_4-7-steiner-定理","aria-label":'Permalink to "4.7: Steiner 定理"'},"​")],-1),z=s("ol",{start:"7"},[s("li",null,[a("Steiner の定理（平行軸の定理）も同じような形で は質量中心の回転軸からの距離）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mo",null,"+"),s("mi",null,"m"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"I=I_c+m b^2 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8641em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])])],-1),k=s("h3",{id:"_4-8-ポイント-6",tabindex:"-1"},[a("4.8: ポイント 6 "),s("a",{class:"header-anchor",href:"#_4-8-ポイント-6","aria-label":'Permalink to "4.8: ポイント 6"'},"​")],-1),_=s("ol",{start:"8"},[s("li",null,[a("ポイント 6 の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{P}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])])])])]),a(" を用いて, Newton の第二法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"P"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}_{\\Sigma}=\\mathrm{d} \\boldsymbol{P} / \\mathrm{d} t, \\boldsymbol{M}_{\\Sigma}=\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8361em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])])])])])])],-1),M=s("h3",{id:"_4-9-ポイント-5",tabindex:"-1"},[a("4.9: ポイント 5 "),s("a",{class:"header-anchor",href:"#_4-9-ポイント-5","aria-label":'Permalink to "4.9: ポイント 5"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("ポイント 5 にに加えて，質量中心を通る "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"z")]),s("annotation",{encoding:"application/x-tex"},"z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")])])]),a(" 軸に対する慣性モ一メントは "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mrow",null,[s("mi",null,"z"),s("mn",null,"0")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"I_{z 0}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",{separator:"true"},","),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"y"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"y"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\sum_{i, j} m_i m_j\\left[\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2\\right] /\\left(2 M_{\\Sigma}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),s("span",{class:"mop"},[s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.162em"}},[s("span",{style:{top:"-2.4003em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4358em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"[")]),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"]")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])],-1),L=s("h3",{id:"_4-10-原点に対する慣性",tabindex:"-1"},[a("4.10: 原点に対する慣性 "),s("a",{class:"header-anchor",href:"#_4-10-原点に対する慣性","aria-label":'Permalink to "4.10: 原点に対する慣性"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("原点に対する慣性モ一メント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"r"),s("mi",null,"i"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\theta=\\sum m_i r_i^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0728em","vertical-align":"-0.2587em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4413em","margin-left":"-0.0278em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2587em"}},[s("span")])])])])])])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"θ"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"y")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"2 \\theta=I_x+I_y+I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9694em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を用いることで二次元物体や等 方性のある物体の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を計算するのに有用.")])],-1),I=s("h3",{id:"_4-11-相当単振子の長",tabindex:"-1"},[a("4.11: 相当単振子の長 "),s("a",{class:"header-anchor",href:"#_4-11-相当単振子の長","aria-label":'Permalink to "4.11: 相当単振子の長"'},"​")],-1),F=s("ol",{start:"11"},[s("li",null,[a("相当単振子の長さが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{l}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9313em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(" である物理振子 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l"),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"−"),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])])]),s("mo",{separator:"true"},","),s("mspace",{width:"1em"}),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"="),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\omega^2(l)=g /\\left(l+I_c / m l\\right) \\\\ & \\omega(l)=\\omega(\\tilde{l}-l)=\\sqrt{g / \\tilde{l}}, \\quad \\tilde{l}=l+I_c / m l \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"5.1641em","vertical-align":"-2.3321em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.7025em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3321em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3745em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])]),s("span",{style:{top:"-3.3345em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as m,c as e,z as s,a,E as t,R as n,o as i,G as p}from"./chunks/framework.a1bac067.js";const gs=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 4","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/4.md","filePath":"academic/physics/ipho-formulas-jpn/4.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/4.md"},c=n('

Formulas for IPhO 日本語版: Section 4 ​

4: 力学 ​

4.1: 剛体の二次元的な平衡 ​

  1. 剛体の二次元的な平衡 : 力についての 2 つの式とトル クについての 1 つの式. 1 （又は 2 ）個の力についての 式は 1（又は 2）個のトルクについての式で代用でき る. トルクの方が良い場合が多く，原点を適切に選択 することで「退屈な」力を消すことができる. もし 2 点 のみに力がかかっているならば，（正味の）力がかかっ ている直線は一致する. 3 点であれば, 3 つの直線は 1 点で交わる.

4.2: 垂直抗力 ​

',5),h=s("ol",{start:"2"},[s("li",null,[a("垂直抗力と摩擦力は 1 つの力に合成でき, 垂直抗力に 対して "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"arctan"),s("mo",null,"⁡"),s("mi",null,"μ")]),s("annotation",{encoding:"application/x-tex"},"\\arctan \\mu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},"arctan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"μ")])])]),a(" の角度で接触点に加わる.")])],-1),o=s("h3",{id:"_4-3-並進運動と回転運動",tabindex:"-1"},[a("4.3: 並進運動と回転運動 "),s("a",{class:"header-anchor",href:"#_4-3-並進運動と回転運動","aria-label":'Permalink to "4.3: 並進運動と回転運動"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("並進運動と回転運動についての Newton の第二法則："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"m"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"ε"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=m \\boldsymbol{a}, \\boldsymbol{M}=I \\boldsymbol{\\varepsilon} \\quad(\\boldsymbol{M}=\\boldsymbol{r} \\times \\boldsymbol{F}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8805em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"ε")])]),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mclose"},")")])])])])]),a(" 二次元の場合には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"M")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε")])])]),a(" は本質的にスカラーで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"l"),s("mo",null,"="),s("msub",null,[s("mi",null,"F"),s("mi",null,"t")]),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"M=F l=F_t r(l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"Fl"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2806em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"t")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" は力のうでの長さ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_4-4-一般化座標",tabindex:"-1"},[a("4.4: 一般化座標 "),s("a",{class:"header-anchor",href:"#_4-4-一般化座標","aria-label":'Permalink to "4.4: 一般化座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("一般化座標. 系の状態が 1 つの変数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"\\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(" とその時間微分 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"˙")])]),s("annotation",{encoding:"application/x-tex"},"\\dot{\\xi}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.0278em"}},[s("span",{class:"mord"},"˙")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(" で表され，ポテンシャルエネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"U=U(\\xi)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")")])])]),a(", 運動エネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"K"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"ξ"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")]),s("annotation",{encoding:"application/x-tex"},"K=\\mu \\xi^2 / 2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2")])])]),a(" であるならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"μ"),s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"¨")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\mu \\ddot{\\xi}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"¨")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"-\\mathrm{d} U(\\xi) / \\mathrm{d} \\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(". (したがって並進運動では, 力はポテン シャルエネルギーの微分)")])],-1),y=s("h3",{id:"_4-5-系質点",tabindex:"-1"},[a("4.5: 系質点 "),s("a",{class:"header-anchor",href:"#_4-5-系質点","aria-label":'Permalink to "4.5: 系質点"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("系が質点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"m"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"m_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" で構成されているとき："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"c")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mi",{mathvariant:"normal"},"/"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"i"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msubsup",null,[s("mi",null,"x"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"y"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"∫"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"y"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"m")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\boldsymbol{r}_c=\\sum m_i \\boldsymbol{r}_i / \\sum m_j, \\boldsymbol{P}=\\sum m_i \\boldsymbol{v}_i \\\\ & \\boldsymbol{L}=\\sum m_i \\boldsymbol{r}_i \\times \\boldsymbol{v}_i, K=\\sum m_i v_i^2 / 2 \\\\ & I_z=\\sum m_i\\left(x_i^2+y_i^2\\right)=\\int\\left(x^2+y^2\\right) \\mathrm{d} m \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8223em","vertical-align":"-3.6611em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_4-6-質量中心の速度",tabindex:"-1"},[a("4.6: 質量中心の速度 "),s("a",{class:"header-anchor",href:"#_4-6-質量中心の速度","aria-label":'Permalink to "4.6: 質量中心の速度"'},"​")],-1),x=s("ol",{start:"6"},[s("li",null,[a("質量中心の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" であるような系 (添え字 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" は質量 中心についての物理量であることを示す）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"R"),s("mi",null,"c")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("msub",null,[s("mi",null,"K"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"c"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},".")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{L}=\\boldsymbol{L}_c+M_{\\Sigma} \\boldsymbol{R}_c \\times \\boldsymbol{v}_c, K=K_c+M_{\\Sigma} v_c^2 / 2 \\\\ \\boldsymbol{P}=\\boldsymbol{P}_c+M_{\\Sigma} \\boldsymbol{v}_c . \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_4-7-steiner-定理",tabindex:"-1"},[a("4.7: Steiner 定理 "),s("a",{class:"header-anchor",href:"#_4-7-steiner-定理","aria-label":'Permalink to "4.7: Steiner 定理"'},"​")],-1),z=s("ol",{start:"7"},[s("li",null,[a("Steiner の定理（平行軸の定理）も同じような形で は質量中心の回転軸からの距離）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mo",null,"+"),s("mi",null,"m"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"I=I_c+m b^2 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8641em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])])],-1),k=s("h3",{id:"_4-8-ポイント-6",tabindex:"-1"},[a("4.8: ポイント 6 "),s("a",{class:"header-anchor",href:"#_4-8-ポイント-6","aria-label":'Permalink to "4.8: ポイント 6"'},"​")],-1),_=s("ol",{start:"8"},[s("li",null,[a("ポイント 6 の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{P}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])])])])]),a(" を用いて, Newton の第二法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"P"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}_{\\Sigma}=\\mathrm{d} \\boldsymbol{P} / \\mathrm{d} t, \\boldsymbol{M}_{\\Sigma}=\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8361em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])])])])])])],-1),M=s("h3",{id:"_4-9-ポイント-5",tabindex:"-1"},[a("4.9: ポイント 5 "),s("a",{class:"header-anchor",href:"#_4-9-ポイント-5","aria-label":'Permalink to "4.9: ポイント 5"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("ポイント 5 にに加えて，質量中心を通る "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"z")]),s("annotation",{encoding:"application/x-tex"},"z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")])])]),a(" 軸に対する慣性モ一メントは "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mrow",null,[s("mi",null,"z"),s("mn",null,"0")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"I_{z 0}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",{separator:"true"},","),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"y"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"y"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\sum_{i, j} m_i m_j\\left[\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2\\right] /\\left(2 M_{\\Sigma}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),s("span",{class:"mop"},[s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.162em"}},[s("span",{style:{top:"-2.4003em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4358em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"[")]),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"]")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])],-1),L=s("h3",{id:"_4-10-原点に対する慣性",tabindex:"-1"},[a("4.10: 原点に対する慣性 "),s("a",{class:"header-anchor",href:"#_4-10-原点に対する慣性","aria-label":'Permalink to "4.10: 原点に対する慣性"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("原点に対する慣性モ一メント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"r"),s("mi",null,"i"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\theta=\\sum m_i r_i^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0728em","vertical-align":"-0.2587em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4413em","margin-left":"-0.0278em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2587em"}},[s("span")])])])])])])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"θ"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"y")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"2 \\theta=I_x+I_y+I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9694em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を用いることで二次元物体や等 方性のある物体の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を計算するのに有用.")])],-1),I=s("h3",{id:"_4-11-相当単振子の長",tabindex:"-1"},[a("4.11: 相当単振子の長 "),s("a",{class:"header-anchor",href:"#_4-11-相当単振子の長","aria-label":'Permalink to "4.11: 相当単振子の長"'},"​")],-1),F=s("ol",{start:"11"},[s("li",null,[a("相当単振子の長さが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{l}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9313em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(" である物理振子 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l"),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"−"),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])])]),s("mo",{separator:"true"},","),s("mspace",{width:"1em"}),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"="),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\omega^2(l)=g /\\left(l+I_c / m l\\right) \\\\ & \\omega(l)=\\omega(\\tilde{l}-l)=\\sqrt{g / \\tilde{l}}, \\quad \\tilde{l}=l+I_c / m l \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"5.1641em","vertical-align":"-2.3321em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.7025em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3321em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3745em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])]),s("span",{style:{top:"-3.3345em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.lean.js b/assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.lean.js rename to assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.lean.js index d2e8a54a..8ca9d301 100644 --- a/assets/academic_physics_ipho-formulas-jpn_4.md.8aeca4b5.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_4.md.6ae94d2b.lean.js @@ -1,4 +1,4 @@ -import{_ as m,c as e,z as s,a,E as t,R as n,o as i,G as p}from"./chunks/framework.a1bac067.js";const gs=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 4","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/4.md","filePath":"academic/physics/ipho-formulas-jpn/4.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/4.md"},c=n("",5),h=s("ol",{start:"2"},[s("li",null,[a("垂直抗力と摩擦力は 1 つの力に合成でき, 垂直抗力に 対して "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"arctan"),s("mo",null,"⁡"),s("mi",null,"μ")]),s("annotation",{encoding:"application/x-tex"},"\\arctan \\mu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},"arctan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"μ")])])]),a(" の角度で接触点に加わる.")])],-1),o=s("h3",{id:"_4-3-並進運動と回転運動",tabindex:"-1"},[a("4.3: 並進運動と回転運動 "),s("a",{class:"header-anchor",href:"#_4-3-並進運動と回転運動","aria-label":'Permalink to "4.3: 並進運動と回転運動"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("並進運動と回転運動についての Newton の第二法則："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"m"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"ε"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=m \\boldsymbol{a}, \\boldsymbol{M}=I \\boldsymbol{\\varepsilon} \\quad(\\boldsymbol{M}=\\boldsymbol{r} \\times \\boldsymbol{F}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8805em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"ε")])]),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mclose"},")")])])])])]),a(" 二次元の場合には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"M")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε")])])]),a(" は本質的にスカラーで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"l"),s("mo",null,"="),s("msub",null,[s("mi",null,"F"),s("mi",null,"t")]),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"M=F l=F_t r(l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"Fl"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2806em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"t")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" は力のうでの長さ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_4-4-一般化座標",tabindex:"-1"},[a("4.4: 一般化座標 "),s("a",{class:"header-anchor",href:"#_4-4-一般化座標","aria-label":'Permalink to "4.4: 一般化座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("一般化座標. 系の状態が 1 つの変数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"\\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(" とその時間微分 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"˙")])]),s("annotation",{encoding:"application/x-tex"},"\\dot{\\xi}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.0278em"}},[s("span",{class:"mord"},"˙")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(" で表され，ポテンシャルエネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"U=U(\\xi)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")")])])]),a(", 運動エネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"K"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"ξ"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")]),s("annotation",{encoding:"application/x-tex"},"K=\\mu \\xi^2 / 2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2")])])]),a(" であるならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"μ"),s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"¨")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\mu \\ddot{\\xi}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"¨")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"-\\mathrm{d} U(\\xi) / \\mathrm{d} \\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(". (したがって並進運動では, 力はポテン シャルエネルギーの微分)")])],-1),y=s("h3",{id:"_4-5-系質点",tabindex:"-1"},[a("4.5: 系質点 "),s("a",{class:"header-anchor",href:"#_4-5-系質点","aria-label":'Permalink to "4.5: 系質点"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("系が質点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"m"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"m_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" で構成されているとき："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"c")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mi",{mathvariant:"normal"},"/"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"i"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msubsup",null,[s("mi",null,"x"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"y"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"∫"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"y"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"m")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\boldsymbol{r}_c=\\sum m_i \\boldsymbol{r}_i / \\sum m_j, \\boldsymbol{P}=\\sum m_i \\boldsymbol{v}_i \\\\ & \\boldsymbol{L}=\\sum m_i \\boldsymbol{r}_i \\times \\boldsymbol{v}_i, K=\\sum m_i v_i^2 / 2 \\\\ & I_z=\\sum m_i\\left(x_i^2+y_i^2\\right)=\\int\\left(x^2+y^2\\right) \\mathrm{d} m \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8223em","vertical-align":"-3.6611em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_4-6-質量中心の速度",tabindex:"-1"},[a("4.6: 質量中心の速度 "),s("a",{class:"header-anchor",href:"#_4-6-質量中心の速度","aria-label":'Permalink to "4.6: 質量中心の速度"'},"​")],-1),x=s("ol",{start:"6"},[s("li",null,[a("質量中心の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" であるような系 (添え字 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" は質量 中心についての物理量であることを示す）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"R"),s("mi",null,"c")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("msub",null,[s("mi",null,"K"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"c"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},".")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{L}=\\boldsymbol{L}_c+M_{\\Sigma} \\boldsymbol{R}_c \\times \\boldsymbol{v}_c, K=K_c+M_{\\Sigma} v_c^2 / 2 \\\\ \\boldsymbol{P}=\\boldsymbol{P}_c+M_{\\Sigma} \\boldsymbol{v}_c . \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_4-7-steiner-定理",tabindex:"-1"},[a("4.7: Steiner 定理 "),s("a",{class:"header-anchor",href:"#_4-7-steiner-定理","aria-label":'Permalink to "4.7: Steiner 定理"'},"​")],-1),z=s("ol",{start:"7"},[s("li",null,[a("Steiner の定理（平行軸の定理）も同じような形で は質量中心の回転軸からの距離）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mo",null,"+"),s("mi",null,"m"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"I=I_c+m b^2 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8641em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])])],-1),k=s("h3",{id:"_4-8-ポイント-6",tabindex:"-1"},[a("4.8: ポイント 6 "),s("a",{class:"header-anchor",href:"#_4-8-ポイント-6","aria-label":'Permalink to "4.8: ポイント 6"'},"​")],-1),_=s("ol",{start:"8"},[s("li",null,[a("ポイント 6 の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{P}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])])])])]),a(" を用いて, Newton の第二法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"P"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}_{\\Sigma}=\\mathrm{d} \\boldsymbol{P} / \\mathrm{d} t, \\boldsymbol{M}_{\\Sigma}=\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8361em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])])])])])])],-1),M=s("h3",{id:"_4-9-ポイント-5",tabindex:"-1"},[a("4.9: ポイント 5 "),s("a",{class:"header-anchor",href:"#_4-9-ポイント-5","aria-label":'Permalink to "4.9: ポイント 5"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("ポイント 5 にに加えて，質量中心を通る "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"z")]),s("annotation",{encoding:"application/x-tex"},"z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")])])]),a(" 軸に対する慣性モ一メントは "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mrow",null,[s("mi",null,"z"),s("mn",null,"0")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"I_{z 0}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",{separator:"true"},","),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"y"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"y"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\sum_{i, j} m_i m_j\\left[\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2\\right] /\\left(2 M_{\\Sigma}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),s("span",{class:"mop"},[s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.162em"}},[s("span",{style:{top:"-2.4003em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4358em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"[")]),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"]")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])],-1),L=s("h3",{id:"_4-10-原点に対する慣性",tabindex:"-1"},[a("4.10: 原点に対する慣性 "),s("a",{class:"header-anchor",href:"#_4-10-原点に対する慣性","aria-label":'Permalink to "4.10: 原点に対する慣性"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("原点に対する慣性モ一メント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"r"),s("mi",null,"i"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\theta=\\sum m_i r_i^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0728em","vertical-align":"-0.2587em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4413em","margin-left":"-0.0278em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2587em"}},[s("span")])])])])])])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"θ"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"y")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"2 \\theta=I_x+I_y+I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9694em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を用いることで二次元物体や等 方性のある物体の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を計算するのに有用.")])],-1),I=s("h3",{id:"_4-11-相当単振子の長",tabindex:"-1"},[a("4.11: 相当単振子の長 "),s("a",{class:"header-anchor",href:"#_4-11-相当単振子の長","aria-label":'Permalink to "4.11: 相当単振子の長"'},"​")],-1),F=s("ol",{start:"11"},[s("li",null,[a("相当単振子の長さが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{l}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9313em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(" である物理振子 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l"),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"−"),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])])]),s("mo",{separator:"true"},","),s("mspace",{width:"1em"}),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"="),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\omega^2(l)=g /\\left(l+I_c / m l\\right) \\\\ & \\omega(l)=\\omega(\\tilde{l}-l)=\\sqrt{g / \\tilde{l}}, \\quad \\tilde{l}=l+I_c / m l \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"5.1641em","vertical-align":"-2.3321em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.7025em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3321em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3745em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])]),s("span",{style:{top:"-3.3345em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as m,c as e,z as s,a,E as t,R as n,o as i,G as p}from"./chunks/framework.a1bac067.js";const gs=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 4","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/4.md","filePath":"academic/physics/ipho-formulas-jpn/4.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/4.md"},c=n("",5),h=s("ol",{start:"2"},[s("li",null,[a("垂直抗力と摩擦力は 1 つの力に合成でき, 垂直抗力に 対して "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"arctan"),s("mo",null,"⁡"),s("mi",null,"μ")]),s("annotation",{encoding:"application/x-tex"},"\\arctan \\mu")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},"arctan"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"μ")])])]),a(" の角度で接触点に加わる.")])],-1),o=s("h3",{id:"_4-3-並進運動と回転運動",tabindex:"-1"},[a("4.3: 並進運動と回転運動 "),s("a",{class:"header-anchor",href:"#_4-3-並進運動と回転運動","aria-label":'Permalink to "4.3: 並進運動と回転運動"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("並進運動と回転運動についての Newton の第二法則："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"m"),s("mi",{mathvariant:"bold-italic"},"a"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"ε"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"M"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=m \\boldsymbol{a}, \\boldsymbol{M}=I \\boldsymbol{\\varepsilon} \\quad(\\boldsymbol{M}=\\boldsymbol{r} \\times \\boldsymbol{F}) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8805em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"a")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"ε")])]),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mclose"},")")])])])])]),a(" 二次元の場合には "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"M")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ε")]),s("annotation",{encoding:"application/x-tex"},"\\varepsilon")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"ε")])])]),a(" は本質的にスカラーで, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"M"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"l"),s("mo",null,"="),s("msub",null,[s("mi",null,"F"),s("mi",null,"t")]),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"M=F l=F_t r(l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"Fl"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2806em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1389em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"t")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(" は力のうでの長さ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_4-4-一般化座標",tabindex:"-1"},[a("4.4: 一般化座標 "),s("a",{class:"header-anchor",href:"#_4-4-一般化座標","aria-label":'Permalink to "4.4: 一般化座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("一般化座標. 系の状態が 1 つの変数 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"\\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(" とその時間微分 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"˙")])]),s("annotation",{encoding:"application/x-tex"},"\\dot{\\xi}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.0278em"}},[s("span",{class:"mord"},"˙")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(" で表され，ポテンシャルエネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"U=U(\\xi)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")")])])]),a(", 運動エネルギーが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"K"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"ξ"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")]),s("annotation",{encoding:"application/x-tex"},"K=\\mu \\xi^2 / 2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/2")])])]),a(" であるならば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"μ"),s("mover",{accent:"true"},[s("mi",null,"ξ"),s("mo",null,"¨")]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\mu \\ddot{\\xi}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1257em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")]),s("span",{style:{top:"-3.2634em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"¨")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"U"),s("mo",{stretchy:"false"},"("),s("mi",null,"ξ"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"ξ")]),s("annotation",{encoding:"application/x-tex"},"-\\mathrm{d} U(\\xi) / \\mathrm{d} \\xi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04601em"}},"ξ")])])]),a(". (したがって並進運動では, 力はポテン シャルエネルギーの微分)")])],-1),y=s("h3",{id:"_4-5-系質点",tabindex:"-1"},[a("4.5: 系質点 "),s("a",{class:"header-anchor",href:"#_4-5-系質点","aria-label":'Permalink to "4.5: 系質点"'},"​")],-1),v=s("ol",{start:"5"},[s("li",null,[a("系が質点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"m"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"m_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" で構成されているとき："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"c")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mi",{mathvariant:"normal"},"/"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"i"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("msubsup",null,[s("mi",null,"x"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"y"),s("mi",null,"i"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mo",null,"∫"),s("mrow",null,[s("mo",{fence:"true"},"("),s("msup",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mi",null,"y"),s("mn",null,"2")]),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"m")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\boldsymbol{r}_c=\\sum m_i \\boldsymbol{r}_i / \\sum m_j, \\boldsymbol{P}=\\sum m_i \\boldsymbol{v}_i \\\\ & \\boldsymbol{L}=\\sum m_i \\boldsymbol{r}_i \\times \\boldsymbol{v}_i, K=\\sum m_i v_i^2 / 2 \\\\ & I_z=\\sum m_i\\left(x_i^2+y_i^2\\right)=\\int\\left(x^2+y^2\\right) \\mathrm{d} m \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"7.8223em","vertical-align":"-3.6611em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"4.1611em"}},[s("span",{style:{top:"-6.4711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-4.5711em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3611em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"m")])]),s("span",{style:{top:"-0.3589em"}},[s("span",{class:"pstrut",style:{height:"3.36em"}}),s("span",{class:"mord"},[s("span",{class:"mord"})])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"3.6611em"}},[s("span")])])])])])])])])])])])])],-1),b=s("h3",{id:"_4-6-質量中心の速度",tabindex:"-1"},[a("4.6: 質量中心の速度 "),s("a",{class:"header-anchor",href:"#_4-6-質量中心の速度","aria-label":'Permalink to "4.6: 質量中心の速度"'},"​")],-1),x=s("ol",{start:"6"},[s("li",null,[a("質量中心の速度が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")])]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{v}_c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5944em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" であるような系 (添え字 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"c")]),s("annotation",{encoding:"application/x-tex"},"c")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c")])])]),a(" は質量 中心についての物理量であることを示す）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"L"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"R"),s("mi",null,"c")]),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mo",{separator:"true"},","),s("mi",null,"K"),s("mo",null,"="),s("msub",null,[s("mi",null,"K"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msubsup",null,[s("mi",null,"v"),s("mi",null,"c"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mn",null,"2")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mo",null,"="),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"P"),s("mi",null,"c")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"v"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},".")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} \\boldsymbol{L}=\\boldsymbol{L}_c+M_{\\Sigma} \\boldsymbol{R}_c \\times \\boldsymbol{v}_c, K=K_c+M_{\\Sigma} v_c^2 / 2 \\\\ \\boldsymbol{P}=\\boldsymbol{P}_c+M_{\\Sigma} \\boldsymbol{v}_c . \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0241em","vertical-align":"-1.2621em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7621em"}},[s("span",{style:{top:"-3.8979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.00421em"}},"R")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"K"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},"/2")])]),s("span",{style:{top:"-2.3979em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2621em"}},[s("span")])])])])])])])])])])])])],-1),w=s("h3",{id:"_4-7-steiner-定理",tabindex:"-1"},[a("4.7: Steiner 定理 "),s("a",{class:"header-anchor",href:"#_4-7-steiner-定理","aria-label":'Permalink to "4.7: Steiner 定理"'},"​")],-1),z=s("ol",{start:"7"},[s("li",null,[a("Steiner の定理（平行軸の定理）も同じような形で は質量中心の回転軸からの距離）:"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mo",null,"+"),s("mi",null,"m"),s("msup",null,[s("mi",null,"b"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"I=I_c+m b^2 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8641em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"b"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])])])],-1),k=s("h3",{id:"_4-8-ポイント-6",tabindex:"-1"},[a("4.8: ポイント 6 "),s("a",{class:"header-anchor",href:"#_4-8-ポイント-6","aria-label":'Permalink to "4.8: ポイント 6"'},"​")],-1),_=s("ol",{start:"8"},[s("li",null,[a("ポイント 6 の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"P")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{P}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"L")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{L}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])])])])]),a(" を用いて, Newton の第二法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"P"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"t")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}_{\\Sigma}=\\mathrm{d} \\boldsymbol{P} / \\mathrm{d} t, \\boldsymbol{M}_{\\Sigma}=\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8361em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"P")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.11424em"}},"M")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"L")])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord mathnormal"},"t")])])])])])])],-1),M=s("h3",{id:"_4-9-ポイント-5",tabindex:"-1"},[a("4.9: ポイント 5 "),s("a",{class:"header-anchor",href:"#_4-9-ポイント-5","aria-label":'Permalink to "4.9: ポイント 5"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("ポイント 5 にに加えて，質量中心を通る "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"z")]),s("annotation",{encoding:"application/x-tex"},"z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.04398em"}},"z")])])]),a(" 軸に対する慣性モ一メントは "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mrow",null,[s("mi",null,"z"),s("mn",null,"0")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"I_{z 0}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z"),s("span",{class:"mord mtight"},"0")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",{separator:"true"},","),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"m"),s("mi",null,"j")]),s("mrow",null,[s("mo",{fence:"true"},"["),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"x"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",null,"+"),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",null,"y"),s("mi",null,"i")]),s("mo",null,"−"),s("msub",null,[s("mi",null,"y"),s("mi",null,"j")]),s("mo",{fence:"true"},")")]),s("mn",null,"2")]),s("mo",{fence:"true"},"]")]),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mn",null,"2"),s("msub",null,[s("mi",null,"M"),s("mi",{mathvariant:"normal"},"Σ")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\sum_{i, j} m_i m_j\\left[\\left(x_i-x_j\\right)^2+\\left(y_i-y_j\\right)^2\\right] /\\left(2 M_{\\Sigma}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),s("span",{class:"mop"},[s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.162em"}},[s("span",{style:{top:"-2.4003em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mpunct mtight"},","),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4358em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"[")]),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05724em"}},"j")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.954em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size2"},"]")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},"2"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"M"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.109em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])])])])],-1),L=s("h3",{id:"_4-10-原点に対する慣性",tabindex:"-1"},[a("4.10: 原点に対する慣性 "),s("a",{class:"header-anchor",href:"#_4-10-原点に対する慣性","aria-label":'Permalink to "4.10: 原点に対する慣性"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("原点に対する慣性モ一メント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ"),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"m"),s("mi",null,"i")]),s("msubsup",null,[s("mi",null,"r"),s("mi",null,"i"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\theta=\\sum m_i r_i^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0728em","vertical-align":"-0.2587em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4413em","margin-left":"-0.0278em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2587em"}},[s("span")])])])])])])])]),a(" は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"θ"),s("mo",null,"="),s("msub",null,[s("mi",null,"I"),s("mi",null,"x")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"y")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"2 \\theta=I_x+I_y+I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9694em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を用いることで二次元物体や等 方性のある物体の "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"z")])]),s("annotation",{encoding:"application/x-tex"},"I_z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.04398em"}},"z")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" を計算するのに有用.")])],-1),I=s("h3",{id:"_4-11-相当単振子の長",tabindex:"-1"},[a("4.11: 相当単振子の長 "),s("a",{class:"header-anchor",href:"#_4-11-相当単振子の長","aria-label":'Permalink to "4.11: 相当単振子の長"'},"​")],-1),F=s("ol",{start:"11"},[s("li",null,[a("相当単振子の長さが "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{l}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9313em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(" である物理振子 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l"),s("mo",{fence:"true"},")")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"ω"),s("mo",{stretchy:"false"},"("),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"−"),s("mi",null,"l"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("msqrt",null,[s("mrow",null,[s("mi",null,"g"),s("mi",{mathvariant:"normal"},"/"),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")])])]),s("mo",{separator:"true"},","),s("mspace",{width:"1em"}),s("mover",{accent:"true"},[s("mi",null,"l"),s("mo",null,"~")]),s("mo",null,"="),s("mi",null,"l"),s("mo",null,"+"),s("msub",null,[s("mi",null,"I"),s("mi",null,"c")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"m"),s("mi",null,"l")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow")])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} & \\omega^2(l)=g /\\left(l+I_c / m l\\right) \\\\ & \\omega(l)=\\omega(\\tilde{l}-l)=\\sqrt{g / \\tilde{l}}, \\quad \\tilde{l}=l+I_c / m l \\\\ & \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"5.1641em","vertical-align":"-2.3321em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})]),s("span",{style:{top:"-1.7025em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"})])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.3321em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.8321em"}},[s("span",{style:{top:"-5.3425em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"c")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")])])]),s("span",{style:{top:"-3.3079em"}},[s("span",{class:"pstrut",style:{height:"3.3745em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mopen"},"("),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3745em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g"),s("span",{class:"mord"},"/"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9313em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")]),s("span",{style:{top:"-3.6134em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1667em"}},[s("span",{class:"mord"},"~")])])])])])])])]),s("span",{style:{top:"-3.3345em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.js b/assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.js rename to assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.js index c77995bd..dd50ccd0 100644 --- a/assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.js +++ b/assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.js @@ -1,4 +1,4 @@ -import{_ as l,c as t,o as n,R as m,z as s,a}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 5","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/5.md","filePath":"academic/physics/ipho-formulas-jpn/5.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/5.md"},i=m('

Formulas for IPhO 日本語版: Section 5 ​

5. 振動と波 ​

5.1: 減衰振動 ​

',3),p=s("ol",null,[s("li",null,[a("減衰振動："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"¨")]),s("mo",null,"+"),s("mn",null,"2"),s("mi",null,"γ"),s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"˙")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mi",null,"γ"),s("mo",null,"<"),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\ddot{x}+2 \\gamma \\dot{x}+\\omega_0^2 x=0(\\gamma<\\omega) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7512em","vertical-align":"-0.0833em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.2222em"}},[s("span",{class:"mord"},"¨")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1111em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.247em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"<"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])])])])]),a(" この方程式の解は ((Section 1: #3)[1#_1-3-定数係数線形微分方程式] 参照) :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"γ"),s("mi",null,"t")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"t"),s("msqrt",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",null,"−"),s("msup",null,[s("mi",null,"γ"),s("mn",null,"2")])])]),s("mo",null,"−"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"x=x_0 e^{-\\gamma t} \\sin \\left(t \\sqrt{\\omega_0^2-\\gamma^2}-\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8436em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"t")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2987em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7959em"}},[s("span",{style:{top:"-2.4337em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.0448em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2663em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.2587em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as l,c as t,o as n,R as m,z as s,a}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 5","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/5.md","filePath":"academic/physics/ipho-formulas-jpn/5.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/5.md"},i=m('

Formulas for IPhO 日本語版: Section 5 ​

5. 振動と波 ​

5.1: 減衰振動 ​

',3),p=s("ol",null,[s("li",null,[a("減衰振動："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"¨")]),s("mo",null,"+"),s("mn",null,"2"),s("mi",null,"γ"),s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"˙")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mi",null,"γ"),s("mo",null,"<"),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\ddot{x}+2 \\gamma \\dot{x}+\\omega_0^2 x=0(\\gamma<\\omega) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7512em","vertical-align":"-0.0833em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.2222em"}},[s("span",{class:"mord"},"¨")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1111em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.247em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"<"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])])])])]),a(" この方程式の解は ((Section 1: #3)[1#_1-3-定数係数線形微分方程式] 参照) :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"γ"),s("mi",null,"t")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"t"),s("msqrt",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",null,"−"),s("msup",null,[s("mi",null,"γ"),s("mn",null,"2")])])]),s("mo",null,"−"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"x=x_0 e^{-\\gamma t} \\sin \\left(t \\sqrt{\\omega_0^2-\\gamma^2}-\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8436em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"t")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2987em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7959em"}},[s("span",{style:{top:"-2.4337em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.0448em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2663em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.2587em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.lean.js b/assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.lean.js rename to assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.lean.js index a35a6cae..d2be5f35 100644 --- a/assets/academic_physics_ipho-formulas-jpn_5.md.d14a555b.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_5.md.db50d685.lean.js @@ -1,4 +1,4 @@ -import{_ as l,c as t,o as n,R as m,z as s,a}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 5","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/5.md","filePath":"academic/physics/ipho-formulas-jpn/5.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/5.md"},i=m("",3),p=s("ol",null,[s("li",null,[a("減衰振動："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"¨")]),s("mo",null,"+"),s("mn",null,"2"),s("mi",null,"γ"),s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"˙")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mi",null,"γ"),s("mo",null,"<"),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\ddot{x}+2 \\gamma \\dot{x}+\\omega_0^2 x=0(\\gamma<\\omega) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7512em","vertical-align":"-0.0833em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.2222em"}},[s("span",{class:"mord"},"¨")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1111em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.247em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"<"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])])])])]),a(" この方程式の解は ((Section 1: #3)[1#_1-3-定数係数線形微分方程式] 参照) :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"γ"),s("mi",null,"t")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"t"),s("msqrt",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",null,"−"),s("msup",null,[s("mi",null,"γ"),s("mn",null,"2")])])]),s("mo",null,"−"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"x=x_0 e^{-\\gamma t} \\sin \\left(t \\sqrt{\\omega_0^2-\\gamma^2}-\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8436em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"t")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2987em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7959em"}},[s("span",{style:{top:"-2.4337em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.0448em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2663em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.2587em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 +import{_ as l,c as t,o as n,R as m,z as s,a}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 5","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/5.md","filePath":"academic/physics/ipho-formulas-jpn/5.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/5.md"},i=m("",3),p=s("ol",null,[s("li",null,[a("減衰振動："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"¨")]),s("mo",null,"+"),s("mn",null,"2"),s("mi",null,"γ"),s("mover",{accent:"true"},[s("mi",null,"x"),s("mo",null,"˙")]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mi",null,"γ"),s("mo",null,"<"),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\ddot{x}+2 \\gamma \\dot{x}+\\omega_0^2 x=0(\\gamma<\\omega) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7512em","vertical-align":"-0.0833em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.2222em"}},[s("span",{class:"mord"},"¨")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"x")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1111em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.247em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"<"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])])])])]),a(" この方程式の解は ((Section 1: #3)[1#_1-3-定数係数線形微分方程式] 参照) :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("msup",null,[s("mi",null,"e"),s("mrow",null,[s("mo",null,"−"),s("mi",null,"γ"),s("mi",null,"t")])]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"t"),s("msqrt",null,[s("mrow",null,[s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mo",null,"−"),s("msup",null,[s("mi",null,"γ"),s("mn",null,"2")])])]),s("mo",null,"−"),s("msub",null,[s("mi",null,"φ"),s("mn",null,"0")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"x=x_0 e^{-\\gamma t} \\sin \\left(t \\sqrt{\\omega_0^2-\\gamma^2}-\\varphi_0\\right) ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"e"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8436em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"mord mathnormal mtight"},"t")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2987em"}},[s("span",{class:"svg-align",style:{top:"-3.8em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7959em"}},[s("span",{style:{top:"-2.4337em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"0")])]),s("span",{style:{top:"-3.0448em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2663em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05556em"}},"γ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.2587em"}},[s("span",{class:"pstrut",style:{height:"3.8em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.88em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.88em",viewBox:"0 0 400000 1944",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 diff --git a/assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.js b/assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.js rename to assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.js index a3c47950..613f16d5 100644 --- a/assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.js +++ b/assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.js @@ -1 +1 @@ -import{_ as t,c as n,o as e,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const D=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 6","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/6.md","filePath":"academic/physics/ipho-formulas-jpn/6.md","lastUpdated":1684858945000}'),m={name:"academic/physics/ipho-formulas-jpn/6.md"},i=l('

Formulas for IPhO 日本語版: Section 6 ​

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

',3),r=s("ol",null,[s("li",null,[a("Fermat の原理 : 点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" から "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" への波の経路は波の移動 時間が最も短いもの.")])],-1),p=s("h3",{id:"_6-2-snell-法則",tabindex:"-1"},[a("6.2: Snell 法則 "),s("a",{class:"header-anchor",href:"#_6-2-snell-法則","aria-label":'Permalink to "6.2: Snell 法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,"Snell の法則 :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"n"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\sin \\alpha_1 / \\sin \\alpha_2=n_2 / n_1=v_1 / v_2 . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])])])],-1),o=s("h3",{id:"_6-3-屈折率",tabindex:"-1"},[a("6.3: 屈折率 "),s("a",{class:"header-anchor",href:"#_6-3-屈折率","aria-label":'Permalink to "6.3: 屈折率"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("屈折率が連続的に変化するならば，媒質を屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"n"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"n"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"n^{\\prime}=n / r \\quad(r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" は曲率半径 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_6-4-屈折率な座標",tabindex:"-1"},[a("6.4: 屈折率な座標 "),s("a",{class:"header-anchor",href:"#_6-4-屈折率な座標","aria-label":'Permalink to "6.4: 屈折率な座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"z")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{z}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathrm"},"z")])])]),a(" 座標にのみ依存するならば, 光子の運動量 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])]),s("annotation",{encoding:"application/x-tex"},"p_x, p_y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" とエネルギーは保存される:")])],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"k"),s("mi",null,"y")]),s("mo",null,"="),s("mtext",null," const., "),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"k"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n"),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"k_x, k_y=\\text { const., }|\\boldsymbol{k}| / n=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const., ")]),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mord"},"∣/"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])],-1),x=s("h3",{id:"_6-5-薄いレンズの式",tabindex:"-1"},[a("6.5:薄いレンズの式 "),s("a",{class:"header-anchor",href:"#_6-5-薄いレンズの式","aria-label":'Permalink to "6.5:薄いレンズの式"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,"薄いレンズの式（符号に注意する）：")],-1),_=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"+"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"b"),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"f"),s("mo",null,"≡"),s("mi",null,"D")]),s("annotation",{encoding:"application/x-tex"},"1 / a+1 / b=1 / f \\equiv D ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"D")])])])])],-1),w=s("h3",{id:"_6-6-newton-の式",tabindex:"-1"},[a("6.6: Newton の式 "),s("a",{class:"header-anchor",href:"#_6-6-newton-の式","aria-label":'Permalink to "6.6: Newton の式"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Newton の式 : 物体側焦点から物体までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"x_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", 像側焦点から像までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" とすると, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"f"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_1 x_2=f^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),k=l('

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

',5),M=s("ol",{start:"9"},[s("li",null,[a("光束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ")]),s("annotation",{encoding:"application/x-tex"},"\\Phi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ")])])]),a(" [単位: lumen "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"lm"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\operatorname{lm})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lm")]),s("span",{class:"mclose"},")")])])]),a("] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},"I=\\Phi / \\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/Ω")])])]),a(". 照度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"lux"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"l"),s("mi",{mathvariant:"normal"},"x")]),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\operatorname{lux}(\\mathrm{lx})]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lux")]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"lx")]),s("span",{class:"mclose"},")]")])])]),a(" は（面に入射する） 面積あたりの光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"S")]),s("annotation",{encoding:"application/x-tex"},"E=\\Phi / S")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S")])])]),a(".")])],-1),f=s("h3",{id:"_6-10-gauss-定理",tabindex:"-1"},[a("6.10: Gauss 定理 "),s("a",{class:"header-anchor",href:"#_6-10-gauss-定理","aria-label":'Permalink to "6.10: Gauss 定理"'},"​")],-1),z=s("ol",{start:"10"},[s("li",null,[a("光束についての Gauss の定理 : 光度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の点光源を囲 む閉曲面を通って外に出る光束は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mn",null,"4"),s("mi",null,"π"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=4 \\pi \\sum I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 光 源が 1 つで距離が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" のとき "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"E=I / r^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),P=s("h3",{id:"_6-11-実験のヒント",tabindex:"-1"},[a("6.11: 実験のヒント "),s("a",{class:"header-anchor",href:"#_6-11-実験のヒント","aria-label":'Permalink to "6.11: 実験のヒント"'},"​")],-1),L=s("ol",{start:"11"},[s("li",null,"実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.")],-1),I=[i,r,p,c,h,o,g,u,d,y,x,v,_,w,b,k,M,f,z,P,L];function S(q,T,A,E,N,V){return e(),n("div",null,I)}const F=t(m,[["render",S]]);export{D as __pageData,F as default}; +import{_ as t,c as n,o as e,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const D=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 6","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/6.md","filePath":"academic/physics/ipho-formulas-jpn/6.md","lastUpdated":1684859551000}'),m={name:"academic/physics/ipho-formulas-jpn/6.md"},i=l('

Formulas for IPhO 日本語版: Section 6 ​

6: 幾何光学，測光 ​

6.1: Fermat 原理 ​

',3),r=s("ol",null,[s("li",null,[a("Fermat の原理 : 点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" から "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" への波の経路は波の移動 時間が最も短いもの.")])],-1),p=s("h3",{id:"_6-2-snell-法則",tabindex:"-1"},[a("6.2: Snell 法則 "),s("a",{class:"header-anchor",href:"#_6-2-snell-法則","aria-label":'Permalink to "6.2: Snell 法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,"Snell の法則 :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"n"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\sin \\alpha_1 / \\sin \\alpha_2=n_2 / n_1=v_1 / v_2 . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])])])],-1),o=s("h3",{id:"_6-3-屈折率",tabindex:"-1"},[a("6.3: 屈折率 "),s("a",{class:"header-anchor",href:"#_6-3-屈折率","aria-label":'Permalink to "6.3: 屈折率"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("屈折率が連続的に変化するならば，媒質を屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"n"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"n"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"n^{\\prime}=n / r \\quad(r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" は曲率半径 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_6-4-屈折率な座標",tabindex:"-1"},[a("6.4: 屈折率な座標 "),s("a",{class:"header-anchor",href:"#_6-4-屈折率な座標","aria-label":'Permalink to "6.4: 屈折率な座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"z")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{z}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathrm"},"z")])])]),a(" 座標にのみ依存するならば, 光子の運動量 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])]),s("annotation",{encoding:"application/x-tex"},"p_x, p_y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" とエネルギーは保存される:")])],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"k"),s("mi",null,"y")]),s("mo",null,"="),s("mtext",null," const., "),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"k"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n"),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"k_x, k_y=\\text { const., }|\\boldsymbol{k}| / n=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const., ")]),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mord"},"∣/"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])],-1),x=s("h3",{id:"_6-5-薄いレンズの式",tabindex:"-1"},[a("6.5:薄いレンズの式 "),s("a",{class:"header-anchor",href:"#_6-5-薄いレンズの式","aria-label":'Permalink to "6.5:薄いレンズの式"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,"薄いレンズの式（符号に注意する）：")],-1),_=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"+"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"b"),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"f"),s("mo",null,"≡"),s("mi",null,"D")]),s("annotation",{encoding:"application/x-tex"},"1 / a+1 / b=1 / f \\equiv D ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"D")])])])])],-1),w=s("h3",{id:"_6-6-newton-の式",tabindex:"-1"},[a("6.6: Newton の式 "),s("a",{class:"header-anchor",href:"#_6-6-newton-の式","aria-label":'Permalink to "6.6: Newton の式"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Newton の式 : 物体側焦点から物体までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"x_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", 像側焦点から像までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" とすると, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"f"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_1 x_2=f^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),k=l('

6.7: 像の位置を求める視差法 ​

  1. 像の位置を求める視差法 : 目の位置と垂直に動かした ときに，鉛筆の先が像に対してずれないような位置を 探す.

6.8: レンズを通る光線の経路の幾何学的な描き方 ​

  1. レンズを通る光線の経路の幾何学的な描き方：a) レン ズの中心を通る光線は屈折しない。b) 光軸に平行な光 線は焦点を通る，c) 屈折後, 初めに平行だった光線どうしは焦点面（焦点を通り光軸に垂直な平面）上で集 まる．d) 平面の像は平面であり，この 2 つの平面はレ ンズの平面上で交わる．

6.9: 光束 ​

',5),M=s("ol",{start:"9"},[s("li",null,[a("光束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ")]),s("annotation",{encoding:"application/x-tex"},"\\Phi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ")])])]),a(" [単位: lumen "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"lm"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\operatorname{lm})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lm")]),s("span",{class:"mclose"},")")])])]),a("] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},"I=\\Phi / \\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/Ω")])])]),a(". 照度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"lux"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"l"),s("mi",{mathvariant:"normal"},"x")]),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\operatorname{lux}(\\mathrm{lx})]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lux")]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"lx")]),s("span",{class:"mclose"},")]")])])]),a(" は（面に入射する） 面積あたりの光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"S")]),s("annotation",{encoding:"application/x-tex"},"E=\\Phi / S")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S")])])]),a(".")])],-1),f=s("h3",{id:"_6-10-gauss-定理",tabindex:"-1"},[a("6.10: Gauss 定理 "),s("a",{class:"header-anchor",href:"#_6-10-gauss-定理","aria-label":'Permalink to "6.10: Gauss 定理"'},"​")],-1),z=s("ol",{start:"10"},[s("li",null,[a("光束についての Gauss の定理 : 光度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の点光源を囲 む閉曲面を通って外に出る光束は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mn",null,"4"),s("mi",null,"π"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=4 \\pi \\sum I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 光 源が 1 つで距離が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" のとき "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"E=I / r^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),P=s("h3",{id:"_6-11-実験のヒント",tabindex:"-1"},[a("6.11: 実験のヒント "),s("a",{class:"header-anchor",href:"#_6-11-実験のヒント","aria-label":'Permalink to "6.11: 実験のヒント"'},"​")],-1),L=s("ol",{start:"11"},[s("li",null,"実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.")],-1),I=[i,r,p,c,h,o,g,u,d,y,x,v,_,w,b,k,M,f,z,P,L];function S(q,T,A,E,N,V){return e(),n("div",null,I)}const F=t(m,[["render",S]]);export{D as __pageData,F as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.lean.js b/assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.lean.js rename to assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.lean.js index 304bcde8..efac9e2b 100644 --- a/assets/academic_physics_ipho-formulas-jpn_6.md.02f25efb.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_6.md.2e655de1.lean.js @@ -1 +1 @@ -import{_ as t,c as n,o as e,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const D=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 6","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/6.md","filePath":"academic/physics/ipho-formulas-jpn/6.md","lastUpdated":1684858945000}'),m={name:"academic/physics/ipho-formulas-jpn/6.md"},i=l("",3),r=s("ol",null,[s("li",null,[a("Fermat の原理 : 点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" から "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" への波の経路は波の移動 時間が最も短いもの.")])],-1),p=s("h3",{id:"_6-2-snell-法則",tabindex:"-1"},[a("6.2: Snell 法則 "),s("a",{class:"header-anchor",href:"#_6-2-snell-法則","aria-label":'Permalink to "6.2: Snell 法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,"Snell の法則 :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"n"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\sin \\alpha_1 / \\sin \\alpha_2=n_2 / n_1=v_1 / v_2 . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])])])],-1),o=s("h3",{id:"_6-3-屈折率",tabindex:"-1"},[a("6.3: 屈折率 "),s("a",{class:"header-anchor",href:"#_6-3-屈折率","aria-label":'Permalink to "6.3: 屈折率"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("屈折率が連続的に変化するならば，媒質を屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"n"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"n"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"n^{\\prime}=n / r \\quad(r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" は曲率半径 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_6-4-屈折率な座標",tabindex:"-1"},[a("6.4: 屈折率な座標 "),s("a",{class:"header-anchor",href:"#_6-4-屈折率な座標","aria-label":'Permalink to "6.4: 屈折率な座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"z")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{z}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathrm"},"z")])])]),a(" 座標にのみ依存するならば, 光子の運動量 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])]),s("annotation",{encoding:"application/x-tex"},"p_x, p_y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" とエネルギーは保存される:")])],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"k"),s("mi",null,"y")]),s("mo",null,"="),s("mtext",null," const., "),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"k"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n"),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"k_x, k_y=\\text { const., }|\\boldsymbol{k}| / n=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const., ")]),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mord"},"∣/"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])],-1),x=s("h3",{id:"_6-5-薄いレンズの式",tabindex:"-1"},[a("6.5:薄いレンズの式 "),s("a",{class:"header-anchor",href:"#_6-5-薄いレンズの式","aria-label":'Permalink to "6.5:薄いレンズの式"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,"薄いレンズの式（符号に注意する）：")],-1),_=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"+"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"b"),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"f"),s("mo",null,"≡"),s("mi",null,"D")]),s("annotation",{encoding:"application/x-tex"},"1 / a+1 / b=1 / f \\equiv D ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"D")])])])])],-1),w=s("h3",{id:"_6-6-newton-の式",tabindex:"-1"},[a("6.6: Newton の式 "),s("a",{class:"header-anchor",href:"#_6-6-newton-の式","aria-label":'Permalink to "6.6: Newton の式"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Newton の式 : 物体側焦点から物体までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"x_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", 像側焦点から像までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" とすると, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"f"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_1 x_2=f^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),k=l("",5),M=s("ol",{start:"9"},[s("li",null,[a("光束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ")]),s("annotation",{encoding:"application/x-tex"},"\\Phi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ")])])]),a(" [単位: lumen "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"lm"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\operatorname{lm})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lm")]),s("span",{class:"mclose"},")")])])]),a("] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},"I=\\Phi / \\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/Ω")])])]),a(". 照度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"lux"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"l"),s("mi",{mathvariant:"normal"},"x")]),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\operatorname{lux}(\\mathrm{lx})]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lux")]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"lx")]),s("span",{class:"mclose"},")]")])])]),a(" は（面に入射する） 面積あたりの光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"S")]),s("annotation",{encoding:"application/x-tex"},"E=\\Phi / S")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S")])])]),a(".")])],-1),f=s("h3",{id:"_6-10-gauss-定理",tabindex:"-1"},[a("6.10: Gauss 定理 "),s("a",{class:"header-anchor",href:"#_6-10-gauss-定理","aria-label":'Permalink to "6.10: Gauss 定理"'},"​")],-1),z=s("ol",{start:"10"},[s("li",null,[a("光束についての Gauss の定理 : 光度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の点光源を囲 む閉曲面を通って外に出る光束は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mn",null,"4"),s("mi",null,"π"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=4 \\pi \\sum I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 光 源が 1 つで距離が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" のとき "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"E=I / r^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),P=s("h3",{id:"_6-11-実験のヒント",tabindex:"-1"},[a("6.11: 実験のヒント "),s("a",{class:"header-anchor",href:"#_6-11-実験のヒント","aria-label":'Permalink to "6.11: 実験のヒント"'},"​")],-1),L=s("ol",{start:"11"},[s("li",null,"実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.")],-1),I=[i,r,p,c,h,o,g,u,d,y,x,v,_,w,b,k,M,f,z,P,L];function S(q,T,A,E,N,V){return e(),n("div",null,I)}const F=t(m,[["render",S]]);export{D as __pageData,F as default}; +import{_ as t,c as n,o as e,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const D=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 6","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/6.md","filePath":"academic/physics/ipho-formulas-jpn/6.md","lastUpdated":1684859551000}'),m={name:"academic/physics/ipho-formulas-jpn/6.md"},i=l("",3),r=s("ol",null,[s("li",null,[a("Fermat の原理 : 点 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"A")]),s("annotation",{encoding:"application/x-tex"},"A")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"A")])])]),a(" から "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" への波の経路は波の移動 時間が最も短いもの.")])],-1),p=s("h3",{id:"_6-2-snell-法則",tabindex:"-1"},[a("6.2: Snell 法則 "),s("a",{class:"header-anchor",href:"#_6-2-snell-法則","aria-label":'Permalink to "6.2: Snell 法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,"Snell の法則 :")],-1),h=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("msub",null,[s("mi",null,"α"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"n"),s("mn",null,"1")]),s("mo",null,"="),s("msub",null,[s("mi",null,"v"),s("mn",null,"1")]),s("mi",{mathvariant:"normal"},"/"),s("msub",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\sin \\alpha_1 / \\sin \\alpha_2=n_2 / n_1=v_1 / v_2 . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0037em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},".")])])])])],-1),o=s("h3",{id:"_6-3-屈折率",tabindex:"-1"},[a("6.3: 屈折率 "),s("a",{class:"header-anchor",href:"#_6-3-屈折率","aria-label":'Permalink to "6.3: 屈折率"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("屈折率が連続的に変化するならば，媒質を屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" で一定のいくつかの仮想的な層に分けて Snell の 法則を適用する. 光線は屈折率一定の層に沿って進む こともでき,もし全反射の条件をわずかに満たせば, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"n"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",null,"="),s("mi",null,"n"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("mspace",{width:"1em"}),s("mo",{stretchy:"false"},"("),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"n^{\\prime}=n / r \\quad(r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" は曲率半径 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},")")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mclose"},")")])])])])],-1),u=s("h3",{id:"_6-4-屈折率な座標",tabindex:"-1"},[a("6.4: 屈折率な座標 "),s("a",{class:"header-anchor",href:"#_6-4-屈折率な座標","aria-label":'Permalink to "6.4: 屈折率な座標"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"z")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{z}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathrm"},"z")])])]),a(" 座標にのみ依存するならば, 光子の運動量 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"p"),s("mi",null,"y")])]),s("annotation",{encoding:"application/x-tex"},"p_x, p_y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])])])])]),a(" とエネルギーは保存される:")])],-1),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"k"),s("mi",null,"x")]),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"k"),s("mi",null,"y")]),s("mo",null,"="),s("mtext",null," const., "),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"bold-italic"},"k"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"n"),s("mo",null,"="),s("mtext",null," const. ")]),s("annotation",{encoding:"application/x-tex"},"k_x, k_y=\\text { const., }|\\boldsymbol{k}| / n=\\text { const. } ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.9805em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"x")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0315em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"y")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const., ")]),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.01852em"}},"k")])]),s("span",{class:"mord"},"∣/"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6151em"}}),s("span",{class:"mord text"},[s("span",{class:"mord"}," const. ")])])])])])],-1),x=s("h3",{id:"_6-5-薄いレンズの式",tabindex:"-1"},[a("6.5:薄いレンズの式 "),s("a",{class:"header-anchor",href:"#_6-5-薄いレンズの式","aria-label":'Permalink to "6.5:薄いレンズの式"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,"薄いレンズの式（符号に注意する）：")],-1),_=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"a"),s("mo",null,"+"),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"b"),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"f"),s("mo",null,"≡"),s("mi",null,"D")]),s("annotation",{encoding:"application/x-tex"},"1 / a+1 / b=1 / f \\equiv D ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal"},"b"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≡"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"D")])])])])],-1),w=s("h3",{id:"_6-6-newton-の式",tabindex:"-1"},[a("6.6: Newton の式 "),s("a",{class:"header-anchor",href:"#_6-6-newton-の式","aria-label":'Permalink to "6.6: Newton の式"'},"​")],-1),b=s("ol",{start:"6"},[s("li",null,[a("Newton の式 : 物体側焦点から物体までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")])]),s("annotation",{encoding:"application/x-tex"},"x_1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(", 像側焦点から像までの距離を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" とすると, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"x"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"x"),s("mn",null,"2")]),s("mo",null,"="),s("msup",null,[s("mi",null,"f"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"x_1 x_2=f^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"x"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),k=l("",5),M=s("ol",{start:"9"},[s("li",null,[a("光束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ")]),s("annotation",{encoding:"application/x-tex"},"\\Phi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ")])])]),a(" [単位: lumen "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"lm"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(\\operatorname{lm})")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lm")]),s("span",{class:"mclose"},")")])])]),a("] は, 光のエネルギー を示し, 眼の感度に応じて重み付けされる. 光度 [candela (cd)]は（光源から出る）立体角あたりの 光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},"I=\\Phi / \\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/Ω")])])]),a(". 照度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"["),s("mi",{mathvariant:"normal"},"lux"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mrow",null,[s("mi",{mathvariant:"normal"},"l"),s("mi",{mathvariant:"normal"},"x")]),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"]")]),s("annotation",{encoding:"application/x-tex"},"[\\operatorname{lux}(\\mathrm{lx})]")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"["),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"lux")]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"lx")]),s("span",{class:"mclose"},")]")])])]),a(" は（面に入射する） 面積あたりの光束で, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"S")]),s("annotation",{encoding:"application/x-tex"},"E=\\Phi / S")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"S")])])]),a(".")])],-1),f=s("h3",{id:"_6-10-gauss-定理",tabindex:"-1"},[a("6.10: Gauss 定理 "),s("a",{class:"header-anchor",href:"#_6-10-gauss-定理","aria-label":'Permalink to "6.10: Gauss 定理"'},"​")],-1),z=s("ol",{start:"10"},[s("li",null,[a("光束についての Gauss の定理 : 光度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" の点光源を囲 む閉曲面を通って外に出る光束は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mn",null,"4"),s("mi",null,"π"),s("mo",null,"∑"),s("msub",null,[s("mi",null,"I"),s("mi",null,"i")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=4 \\pi \\sum I_i")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(". 光 源が 1 つで距離が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" のとき "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"E=I / r^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])])])],-1),P=s("h3",{id:"_6-11-実験のヒント",tabindex:"-1"},[a("6.11: 実験のヒント "),s("a",{class:"header-anchor",href:"#_6-11-実験のヒント","aria-label":'Permalink to "6.11: 実験のヒント"'},"​")],-1),L=s("ol",{start:"11"},[s("li",null,"実験のヒント：紙についた油污れが周囲の紙と同じ明 るさならば，その紙は両面から同じように照らされて いる.")],-1),I=[i,r,p,c,h,o,g,u,d,y,x,v,_,w,b,k,M,f,z,P,L];function S(q,T,A,E,N,V){return e(),n("div",null,I)}const F=t(m,[["render",S]]);export{D as __pageData,F as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.js b/assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.js rename to assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.js index 8ee01b61..e8d7d9d1 100644 --- a/assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.js +++ b/assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,z as s,a,E as e,R as m,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 7","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/7.md","filePath":"academic/physics/ipho-formulas-jpn/7.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/7.md"},c=m('

Formulas for IPhO 日本語版: Section 7 ​

7: 波動光学 ​

7.1: Huygens の原理に基づいた回折 ​

  1. Huygens の原理に基づいた回折 : 障害物が波面を切断 すると波面は小さな断片に分割され，それが仮想的な 点波源となり，観測点での波の振幅はこれらの波源か らの寄与の重ね合わせとなる.

7.2: 二重スリット ​

',5),h=s("ol",{start:"2"},[s("li",null,[a("二重スリット（幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"≪"),s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"d \\ll a, \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7335em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])]),a(" による干渉:強 め合う角 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mi",null,"max"),s("mo",null,"⁡")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈"),s("mi",{mathvariant:"double-struck"},"Z"),s("mi",{mathvariant:"normal"},"."),s("mi",null,"I"),s("mo",null,"∝")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\max }=\\arcsin (n \\lambda / d), n \\in \\mathbb{Z} . I \\propto")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mop mtight"},[s("span",{class:"mtight"},"m"),s("span",{class:"mtight"},"a"),s("span",{class:"mtight"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mord"},"."),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"π"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\cos ^2\\left(k \\frac{a}{2} \\sin \\varphi\\right),(k=2 \\pi / \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])])])],-1),o=s("h3",{id:"_7-3-単スリット-弱め合う角",tabindex:"-1"},[a("7.3: 単スリット-弱め合う角 "),s("a",{class:"header-anchor",href:"#_7-3-単スリット-弱め合う角","aria-label":'Permalink to "7.3: 単スリット-弱め合う角"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("単スリット:弱め合う角: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mtext",null,"min ")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\text {min }}=\\arcsin (n \\lambda / d), n \\in")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3175em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"min ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"double-struck"},"Z"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",{mathvariant:"normal"},"≠"),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\mathbb{Z}, n \\neq 0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mrel"},[s("span",{class:"mord vbox"},[s("span",{class:"thinbox"},[s("span",{class:"rlap"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"inner"},[s("span",{class:"mord"},[s("span",{class:"mrel"},"")])]),s("span",{class:"fix"})])])])]),s("span",{class:"mrel"},"=")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 中央の強め合う部分は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mo",null,"±"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"n=\\pm 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"±"),s("span",{class:"mord"},"1")])])]),a(" の間である ことに注意せよ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"∝"),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"d"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")]),s("annotation",{encoding:"application/x-tex"},"I \\propto \\sin ^2\\left(k \\frac{d}{2} \\sin \\varphi\\right) / \\sin \\varphi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2301em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])])])])],-1),u=s("h3",{id:"_7-4-回折格子",tabindex:"-1"},[a("7.4: 回折格子 "),s("a",{class:"header-anchor",href:"#_7-4-回折格子","aria-label":'Permalink to "7.4: 回折格子"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("回折格子：主な強め合う角はポイント 2 と同じで, 主 な強め合う角の幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" を回折格子の正味の長さとすれ ばポイント 3 と同じ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" 番目の明線のスペクトルの分 解能は，溝の総数を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(" 本として "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=n N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(".")])],-1),y=s("h3",{id:"_7-5-分光器の分解能",tabindex:"-1"},[a("7.5: 分光器の分解能 "),s("a",{class:"header-anchor",href:"#_7-5-分光器の分解能","aria-label":'Permalink to "7.5: 分光器の分解能"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[a("分光器の分解能 : 最短の光線と最長の光線の光学距離 の差を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L")]),s("annotation",{encoding:"application/x-tex"},"L")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L")])])]),a(" として, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mfrac",null,[s("mi",null,"L"),s("mi",null,"λ")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=\\frac{L}{\\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),v=s("h3",{id:"_7-6-プリズムの分解能",tabindex:"-1"},[a("7.6: プリズムの分解能 "),s("a",{class:"header-anchor",href:"#_7-6-プリズムの分解能","aria-label":'Permalink to "7.6: プリズムの分解能"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("プリズムの分解能 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"a"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"n")]),s("mrow",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"λ")])])]),s("annotation",{encoding:"application/x-tex"},": \\frac{\\lambda}{\\Delta \\lambda}=a \\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mspace nobreak mtight"},[s("span",{class:"mtight"}," ")]),s("span",{class:"mord mathrm mtight"},"d")]),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"d"),s("span",{class:"mord mathnormal mtight"},"n")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("h3",{id:"_7-7-角度距離",tabindex:"-1"},[a("7.7: 角度距離 "),s("a",{class:"header-anchor",href:"#_7-7-角度距離","aria-label":'Permalink to "7.7: 角度距離"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("理想的な望遠鏡 (レンズ) で 2 点を解像するときの角度距離 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"≈"),s("mn",null,"1.22"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"\\varphi \\approx 1.22 \\lambda / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.22"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(". この角度では, 一方の点の中 心が他方の点の最初の回折最小值に当たる.")])],-1),_=s("h3",{id:"_7-8-bragg-の法則",tabindex:"-1"},[a("7.8: Bragg の法則 "),s("a",{class:"header-anchor",href:"#_7-8-bragg-の法則","aria-label":'Permalink to "7.8: Bragg の法則"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("Bragg の法則：間隔が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" の平行な結晶面の組は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"d"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"λ")]),s("annotation",{encoding:"application/x-tex"},"2 d \\sin \\theta=n \\lambda")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"nλ")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"X")])])]),a(" 線を反射する. ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ")]),s("annotation",{encoding:"application/x-tex"},"\\theta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ")])])]),a(" は結 晶面と X 線がなす角 (かすめ角).")])],-1),f=s("h3",{id:"_7-9-高密度電体媒質反射",tabindex:"-1"},[a("7.9: 高密度電体媒質反射 "),s("a",{class:"header-anchor",href:"#_7-9-高密度電体媒質反射","aria-label":'Permalink to "7.9: 高密度電体媒質反射"'},"​")],-1),M=s("ol",{start:"9"},[s("li",null,[a("光学的に高密度な誘電体媒質による反射 : 位相が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(" ず れる. 半透明の薄膜では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")]),s("mo",null,"="),s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}+\\phi_{\\leftarrow}=\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(". ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\leftarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は反射波と透過波の位相差（矢印は入射方向を 示す)")])],-1),L=s("h3",{id:"_7-10-fabry-perot-干渉計",tabindex:"-1"},[a("7.10: Fabry-Pérot 干渉計 "),s("a",{class:"header-anchor",href:"#_7-10-fabry-perot-干渉計","aria-label":'Permalink to "7.10: Fabry-Pérot 干渉計"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("Fabry-Pérot 干渉計 : 高い反射率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",null,"≪"),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"r(1-r \\ll 1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" を持 つ 2 枚の平行な半透明の鏡. 分解能は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"ν"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ν")])]),s("mo",null,"≈"),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"a")]),s("mrow",null,[s("mi",null,"λ"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\nu}{\\Delta \\nu} \\approx \\frac{2 a}{\\lambda(1-r)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3651em","vertical-align":"-0.52em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ"),s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mtight"},"1"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose mtight"},")")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.52em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 5 つの平面波 (干渉計の前で左右に進む波, 内部を左右 に進む波，後ろを進む波）を設定して境界条件を課す ことで，透過スペクトルを求められる.")])],-1),B=s("h3",{id:"_7-11-コヒーレントな電磁波",tabindex:"-1"},[a("7.11: コヒーレントな電磁波 "),s("a",{class:"header-anchor",href:"#_7-11-コヒーレントな電磁波","aria-label":'Permalink to "7.11: コヒーレントな電磁波"'},"​")],-1),I=s("ol",{start:"11"},[s("li",null,[a("コヒーレントな電磁波: 電場をベクトル为で表し, ベク トル間の角度を位相差とする. 屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mi",null,"n"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"n=n(\\omega)=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{\\varepsilon(\\omega)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as n,z as s,a,E as e,R as m,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 7","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/7.md","filePath":"academic/physics/ipho-formulas-jpn/7.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/7.md"},c=m('

Formulas for IPhO 日本語版: Section 7 ​

7: 波動光学 ​

7.1: Huygens の原理に基づいた回折 ​

  1. Huygens の原理に基づいた回折 : 障害物が波面を切断 すると波面は小さな断片に分割され，それが仮想的な 点波源となり，観測点での波の振幅はこれらの波源か らの寄与の重ね合わせとなる.

7.2: 二重スリット ​

',5),h=s("ol",{start:"2"},[s("li",null,[a("二重スリット（幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"≪"),s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"d \\ll a, \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7335em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])]),a(" による干渉:強 め合う角 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mi",null,"max"),s("mo",null,"⁡")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈"),s("mi",{mathvariant:"double-struck"},"Z"),s("mi",{mathvariant:"normal"},"."),s("mi",null,"I"),s("mo",null,"∝")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\max }=\\arcsin (n \\lambda / d), n \\in \\mathbb{Z} . I \\propto")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mop mtight"},[s("span",{class:"mtight"},"m"),s("span",{class:"mtight"},"a"),s("span",{class:"mtight"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mord"},"."),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"π"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\cos ^2\\left(k \\frac{a}{2} \\sin \\varphi\\right),(k=2 \\pi / \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])])])],-1),o=s("h3",{id:"_7-3-単スリット-弱め合う角",tabindex:"-1"},[a("7.3: 単スリット-弱め合う角 "),s("a",{class:"header-anchor",href:"#_7-3-単スリット-弱め合う角","aria-label":'Permalink to "7.3: 単スリット-弱め合う角"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("単スリット:弱め合う角: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mtext",null,"min ")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\text {min }}=\\arcsin (n \\lambda / d), n \\in")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3175em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"min ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"double-struck"},"Z"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",{mathvariant:"normal"},"≠"),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\mathbb{Z}, n \\neq 0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mrel"},[s("span",{class:"mord vbox"},[s("span",{class:"thinbox"},[s("span",{class:"rlap"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"inner"},[s("span",{class:"mord"},[s("span",{class:"mrel"},"")])]),s("span",{class:"fix"})])])])]),s("span",{class:"mrel"},"=")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 中央の強め合う部分は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mo",null,"±"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"n=\\pm 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"±"),s("span",{class:"mord"},"1")])])]),a(" の間である ことに注意せよ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"∝"),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"d"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")]),s("annotation",{encoding:"application/x-tex"},"I \\propto \\sin ^2\\left(k \\frac{d}{2} \\sin \\varphi\\right) / \\sin \\varphi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2301em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])])])])],-1),u=s("h3",{id:"_7-4-回折格子",tabindex:"-1"},[a("7.4: 回折格子 "),s("a",{class:"header-anchor",href:"#_7-4-回折格子","aria-label":'Permalink to "7.4: 回折格子"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("回折格子：主な強め合う角はポイント 2 と同じで, 主 な強め合う角の幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" を回折格子の正味の長さとすれ ばポイント 3 と同じ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" 番目の明線のスペクトルの分 解能は，溝の総数を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(" 本として "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=n N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(".")])],-1),y=s("h3",{id:"_7-5-分光器の分解能",tabindex:"-1"},[a("7.5: 分光器の分解能 "),s("a",{class:"header-anchor",href:"#_7-5-分光器の分解能","aria-label":'Permalink to "7.5: 分光器の分解能"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[a("分光器の分解能 : 最短の光線と最長の光線の光学距離 の差を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L")]),s("annotation",{encoding:"application/x-tex"},"L")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L")])])]),a(" として, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mfrac",null,[s("mi",null,"L"),s("mi",null,"λ")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=\\frac{L}{\\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),v=s("h3",{id:"_7-6-プリズムの分解能",tabindex:"-1"},[a("7.6: プリズムの分解能 "),s("a",{class:"header-anchor",href:"#_7-6-プリズムの分解能","aria-label":'Permalink to "7.6: プリズムの分解能"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("プリズムの分解能 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"a"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"n")]),s("mrow",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"λ")])])]),s("annotation",{encoding:"application/x-tex"},": \\frac{\\lambda}{\\Delta \\lambda}=a \\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mspace nobreak mtight"},[s("span",{class:"mtight"}," ")]),s("span",{class:"mord mathrm mtight"},"d")]),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"d"),s("span",{class:"mord mathnormal mtight"},"n")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("h3",{id:"_7-7-角度距離",tabindex:"-1"},[a("7.7: 角度距離 "),s("a",{class:"header-anchor",href:"#_7-7-角度距離","aria-label":'Permalink to "7.7: 角度距離"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("理想的な望遠鏡 (レンズ) で 2 点を解像するときの角度距離 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"≈"),s("mn",null,"1.22"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"\\varphi \\approx 1.22 \\lambda / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.22"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(". この角度では, 一方の点の中 心が他方の点の最初の回折最小值に当たる.")])],-1),_=s("h3",{id:"_7-8-bragg-の法則",tabindex:"-1"},[a("7.8: Bragg の法則 "),s("a",{class:"header-anchor",href:"#_7-8-bragg-の法則","aria-label":'Permalink to "7.8: Bragg の法則"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("Bragg の法則：間隔が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" の平行な結晶面の組は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"d"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"λ")]),s("annotation",{encoding:"application/x-tex"},"2 d \\sin \\theta=n \\lambda")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"nλ")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"X")])])]),a(" 線を反射する. ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ")]),s("annotation",{encoding:"application/x-tex"},"\\theta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ")])])]),a(" は結 晶面と X 線がなす角 (かすめ角).")])],-1),f=s("h3",{id:"_7-9-高密度電体媒質反射",tabindex:"-1"},[a("7.9: 高密度電体媒質反射 "),s("a",{class:"header-anchor",href:"#_7-9-高密度電体媒質反射","aria-label":'Permalink to "7.9: 高密度電体媒質反射"'},"​")],-1),M=s("ol",{start:"9"},[s("li",null,[a("光学的に高密度な誘電体媒質による反射 : 位相が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(" ず れる. 半透明の薄膜では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")]),s("mo",null,"="),s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}+\\phi_{\\leftarrow}=\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(". ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\leftarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は反射波と透過波の位相差（矢印は入射方向を 示す)")])],-1),L=s("h3",{id:"_7-10-fabry-perot-干渉計",tabindex:"-1"},[a("7.10: Fabry-Pérot 干渉計 "),s("a",{class:"header-anchor",href:"#_7-10-fabry-perot-干渉計","aria-label":'Permalink to "7.10: Fabry-Pérot 干渉計"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("Fabry-Pérot 干渉計 : 高い反射率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",null,"≪"),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"r(1-r \\ll 1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" を持 つ 2 枚の平行な半透明の鏡. 分解能は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"ν"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ν")])]),s("mo",null,"≈"),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"a")]),s("mrow",null,[s("mi",null,"λ"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\nu}{\\Delta \\nu} \\approx \\frac{2 a}{\\lambda(1-r)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3651em","vertical-align":"-0.52em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ"),s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mtight"},"1"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose mtight"},")")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.52em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 5 つの平面波 (干渉計の前で左右に進む波, 内部を左右 に進む波，後ろを進む波）を設定して境界条件を課す ことで，透過スペクトルを求められる.")])],-1),B=s("h3",{id:"_7-11-コヒーレントな電磁波",tabindex:"-1"},[a("7.11: コヒーレントな電磁波 "),s("a",{class:"header-anchor",href:"#_7-11-コヒーレントな電磁波","aria-label":'Permalink to "7.11: コヒーレントな電磁波"'},"​")],-1),I=s("ol",{start:"11"},[s("li",null,[a("コヒーレントな電磁波: 電場をベクトル为で表し, ベク トル間の角度を位相差とする. 屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mi",null,"n"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"n=n(\\omega)=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{\\varepsilon(\\omega)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.lean.js b/assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.lean.js rename to assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.lean.js index b6111a5d..05463c2d 100644 --- a/assets/academic_physics_ipho-formulas-jpn_7.md.95da3e92.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_7.md.7be7b512.lean.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,z as s,a,E as e,R as m,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 7","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/7.md","filePath":"academic/physics/ipho-formulas-jpn/7.md","lastUpdated":1684858945000}'),r={name:"academic/physics/ipho-formulas-jpn/7.md"},c=m("",5),h=s("ol",{start:"2"},[s("li",null,[a("二重スリット（幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"≪"),s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"d \\ll a, \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7335em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])]),a(" による干渉:強 め合う角 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mi",null,"max"),s("mo",null,"⁡")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈"),s("mi",{mathvariant:"double-struck"},"Z"),s("mi",{mathvariant:"normal"},"."),s("mi",null,"I"),s("mo",null,"∝")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\max }=\\arcsin (n \\lambda / d), n \\in \\mathbb{Z} . I \\propto")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mop mtight"},[s("span",{class:"mtight"},"m"),s("span",{class:"mtight"},"a"),s("span",{class:"mtight"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mord"},"."),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"π"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\cos ^2\\left(k \\frac{a}{2} \\sin \\varphi\\right),(k=2 \\pi / \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])])])],-1),o=s("h3",{id:"_7-3-単スリット-弱め合う角",tabindex:"-1"},[a("7.3: 単スリット-弱め合う角 "),s("a",{class:"header-anchor",href:"#_7-3-単スリット-弱め合う角","aria-label":'Permalink to "7.3: 単スリット-弱め合う角"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("単スリット:弱め合う角: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mtext",null,"min ")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\text {min }}=\\arcsin (n \\lambda / d), n \\in")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3175em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"min ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"double-struck"},"Z"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",{mathvariant:"normal"},"≠"),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\mathbb{Z}, n \\neq 0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mrel"},[s("span",{class:"mord vbox"},[s("span",{class:"thinbox"},[s("span",{class:"rlap"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"inner"},[s("span",{class:"mord"},[s("span",{class:"mrel"},"")])]),s("span",{class:"fix"})])])])]),s("span",{class:"mrel"},"=")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 中央の強め合う部分は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mo",null,"±"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"n=\\pm 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"±"),s("span",{class:"mord"},"1")])])]),a(" の間である ことに注意せよ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"∝"),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"d"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")]),s("annotation",{encoding:"application/x-tex"},"I \\propto \\sin ^2\\left(k \\frac{d}{2} \\sin \\varphi\\right) / \\sin \\varphi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2301em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])])])])],-1),u=s("h3",{id:"_7-4-回折格子",tabindex:"-1"},[a("7.4: 回折格子 "),s("a",{class:"header-anchor",href:"#_7-4-回折格子","aria-label":'Permalink to "7.4: 回折格子"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("回折格子：主な強め合う角はポイント 2 と同じで, 主 な強め合う角の幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" を回折格子の正味の長さとすれ ばポイント 3 と同じ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" 番目の明線のスペクトルの分 解能は，溝の総数を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(" 本として "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=n N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(".")])],-1),y=s("h3",{id:"_7-5-分光器の分解能",tabindex:"-1"},[a("7.5: 分光器の分解能 "),s("a",{class:"header-anchor",href:"#_7-5-分光器の分解能","aria-label":'Permalink to "7.5: 分光器の分解能"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[a("分光器の分解能 : 最短の光線と最長の光線の光学距離 の差を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L")]),s("annotation",{encoding:"application/x-tex"},"L")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L")])])]),a(" として, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mfrac",null,[s("mi",null,"L"),s("mi",null,"λ")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=\\frac{L}{\\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),v=s("h3",{id:"_7-6-プリズムの分解能",tabindex:"-1"},[a("7.6: プリズムの分解能 "),s("a",{class:"header-anchor",href:"#_7-6-プリズムの分解能","aria-label":'Permalink to "7.6: プリズムの分解能"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("プリズムの分解能 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"a"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"n")]),s("mrow",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"λ")])])]),s("annotation",{encoding:"application/x-tex"},": \\frac{\\lambda}{\\Delta \\lambda}=a \\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mspace nobreak mtight"},[s("span",{class:"mtight"}," ")]),s("span",{class:"mord mathrm mtight"},"d")]),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"d"),s("span",{class:"mord mathnormal mtight"},"n")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("h3",{id:"_7-7-角度距離",tabindex:"-1"},[a("7.7: 角度距離 "),s("a",{class:"header-anchor",href:"#_7-7-角度距離","aria-label":'Permalink to "7.7: 角度距離"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("理想的な望遠鏡 (レンズ) で 2 点を解像するときの角度距離 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"≈"),s("mn",null,"1.22"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"\\varphi \\approx 1.22 \\lambda / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.22"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(". この角度では, 一方の点の中 心が他方の点の最初の回折最小值に当たる.")])],-1),_=s("h3",{id:"_7-8-bragg-の法則",tabindex:"-1"},[a("7.8: Bragg の法則 "),s("a",{class:"header-anchor",href:"#_7-8-bragg-の法則","aria-label":'Permalink to "7.8: Bragg の法則"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("Bragg の法則：間隔が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" の平行な結晶面の組は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"d"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"λ")]),s("annotation",{encoding:"application/x-tex"},"2 d \\sin \\theta=n \\lambda")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"nλ")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"X")])])]),a(" 線を反射する. ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ")]),s("annotation",{encoding:"application/x-tex"},"\\theta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ")])])]),a(" は結 晶面と X 線がなす角 (かすめ角).")])],-1),f=s("h3",{id:"_7-9-高密度電体媒質反射",tabindex:"-1"},[a("7.9: 高密度電体媒質反射 "),s("a",{class:"header-anchor",href:"#_7-9-高密度電体媒質反射","aria-label":'Permalink to "7.9: 高密度電体媒質反射"'},"​")],-1),M=s("ol",{start:"9"},[s("li",null,[a("光学的に高密度な誘電体媒質による反射 : 位相が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(" ず れる. 半透明の薄膜では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")]),s("mo",null,"="),s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}+\\phi_{\\leftarrow}=\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(". ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\leftarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は反射波と透過波の位相差（矢印は入射方向を 示す)")])],-1),L=s("h3",{id:"_7-10-fabry-perot-干渉計",tabindex:"-1"},[a("7.10: Fabry-Pérot 干渉計 "),s("a",{class:"header-anchor",href:"#_7-10-fabry-perot-干渉計","aria-label":'Permalink to "7.10: Fabry-Pérot 干渉計"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("Fabry-Pérot 干渉計 : 高い反射率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",null,"≪"),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"r(1-r \\ll 1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" を持 つ 2 枚の平行な半透明の鏡. 分解能は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"ν"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ν")])]),s("mo",null,"≈"),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"a")]),s("mrow",null,[s("mi",null,"λ"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\nu}{\\Delta \\nu} \\approx \\frac{2 a}{\\lambda(1-r)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3651em","vertical-align":"-0.52em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ"),s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mtight"},"1"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose mtight"},")")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.52em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 5 つの平面波 (干渉計の前で左右に進む波, 内部を左右 に進む波，後ろを進む波）を設定して境界条件を課す ことで，透過スペクトルを求められる.")])],-1),B=s("h3",{id:"_7-11-コヒーレントな電磁波",tabindex:"-1"},[a("7.11: コヒーレントな電磁波 "),s("a",{class:"header-anchor",href:"#_7-11-コヒーレントな電磁波","aria-label":'Permalink to "7.11: コヒーレントな電磁波"'},"​")],-1),I=s("ol",{start:"11"},[s("li",null,[a("コヒーレントな電磁波: 電場をベクトル为で表し, ベク トル間の角度を位相差とする. 屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mi",null,"n"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"n=n(\\omega)=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{\\varepsilon(\\omega)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 +import{_ as t,c as n,z as s,a,E as e,R as m,o as i,G as p}from"./chunks/framework.a1bac067.js";const U=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 7","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/7.md","filePath":"academic/physics/ipho-formulas-jpn/7.md","lastUpdated":1684859551000}'),r={name:"academic/physics/ipho-formulas-jpn/7.md"},c=m("",5),h=s("ol",{start:"2"},[s("li",null,[a("二重スリット（幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"≪"),s("mi",null,"a"),s("mo",{separator:"true"},","),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"d \\ll a, \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7335em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])]),a(" による干渉:強 め合う角 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mi",null,"max"),s("mo",null,"⁡")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈"),s("mi",{mathvariant:"double-struck"},"Z"),s("mi",{mathvariant:"normal"},"."),s("mi",null,"I"),s("mo",null,"∝")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\max }=\\arcsin (n \\lambda / d), n \\in \\mathbb{Z} . I \\propto")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mop mtight"},[s("span",{class:"mtight"},"m"),s("span",{class:"mtight"},"a"),s("span",{class:"mtight"},"x")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6889em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mord"},"."),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mrow",null,[s("mi",null,"cos"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"a"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mo",{separator:"true"},","),s("mo",{stretchy:"false"},"("),s("mi",null,"k"),s("mo",null,"="),s("mn",null,"2"),s("mi",null,"π"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"λ"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"\\cos ^2\\left(k \\frac{a}{2} \\sin \\varphi\\right),(k=2 \\pi / \\lambda)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"cos"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mclose"},")")])])])])],-1),o=s("h3",{id:"_7-3-単スリット-弱め合う角",tabindex:"-1"},[a("7.3: 単スリット-弱め合う角 "),s("a",{class:"header-anchor",href:"#_7-3-単スリット-弱め合う角","aria-label":'Permalink to "7.3: 単スリット-弱め合う角"'},"​")],-1),g=s("ol",{start:"3"},[s("li",null,[a("単スリット:弱め合う角: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"φ"),s("mtext",null,"min ")]),s("mo",null,"="),s("mi",null,"arcsin"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"n"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",null,"∈")]),s("annotation",{encoding:"application/x-tex"},"\\varphi_{\\text {min }}=\\arcsin (n \\lambda / d), n \\in")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3175em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"min ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mop"},"arcsin"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"nλ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∈")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"double-struck"},"Z"),s("mo",{separator:"true"},","),s("mi",null,"n"),s("mo",{mathvariant:"normal"},"≠"),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\mathbb{Z}, n \\neq 0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathbb"},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},[s("span",{class:"mrel"},[s("span",{class:"mord vbox"},[s("span",{class:"thinbox"},[s("span",{class:"rlap"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"inner"},[s("span",{class:"mord"},[s("span",{class:"mrel"},"")])]),s("span",{class:"fix"})])])])]),s("span",{class:"mrel"},"=")]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 中央の強め合う部分は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mo",null,"±"),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"n=\\pm 1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"±"),s("span",{class:"mord"},"1")])])]),a(" の間である ことに注意せよ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I"),s("mo",null,"∝"),s("msup",null,[s("mrow",null,[s("mi",null,"sin"),s("mo",null,"⁡")]),s("mn",null,"2")]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"k"),s("mfrac",null,[s("mi",null,"d"),s("mn",null,"2")]),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ"),s("mo",{fence:"true"},")")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"φ")]),s("annotation",{encoding:"application/x-tex"},"I \\propto \\sin ^2\\left(k \\frac{d}{2} \\sin \\varphi\\right) / \\sin \\varphi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2301em","vertical-align":"-0.35em"}}),s("span",{class:"mop"},[s("span",{class:"mop"},"sin"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8719em"}},[s("span",{style:{top:"-3.1208em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},"/"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"φ")])])])])],-1),u=s("h3",{id:"_7-4-回折格子",tabindex:"-1"},[a("7.4: 回折格子 "),s("a",{class:"header-anchor",href:"#_7-4-回折格子","aria-label":'Permalink to "7.4: 回折格子"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("回折格子：主な強め合う角はポイント 2 と同じで, 主 な強め合う角の幅は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" を回折格子の正味の長さとすれ ばポイント 3 と同じ. "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"n")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])]),a(" 番目の明線のスペクトルの分 解能は，溝の総数を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(" 本として "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"N")]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=n N")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"N")])])]),a(".")])],-1),y=s("h3",{id:"_7-5-分光器の分解能",tabindex:"-1"},[a("7.5: 分光器の分解能 "),s("a",{class:"header-anchor",href:"#_7-5-分光器の分解能","aria-label":'Permalink to "7.5: 分光器の分解能"'},"​")],-1),x=s("ol",{start:"5"},[s("li",null,[a("分光器の分解能 : 最短の光線と最長の光線の光学距離 の差を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L")]),s("annotation",{encoding:"application/x-tex"},"L")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L")])])]),a(" として, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mfrac",null,[s("mi",null,"L"),s("mi",null,"λ")])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\lambda}{\\Delta \\lambda}=\\frac{L}{\\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),v=s("h3",{id:"_7-6-プリズムの分解能",tabindex:"-1"},[a("7.6: プリズムの分解能 "),s("a",{class:"header-anchor",href:"#_7-6-プリズムの分解能","aria-label":'Permalink to "7.6: プリズムの分解能"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("プリズムの分解能 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mfrac",null,[s("mi",null,"λ"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"λ")])]),s("mo",null,"="),s("mi",null,"a"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",null,"n")]),s("mrow",null,[s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"λ")])])]),s("annotation",{encoding:"application/x-tex"},": \\frac{\\lambda}{\\Delta \\lambda}=a \\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2251em","vertical-align":"-0.345em"}}),s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8801em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mspace nobreak mtight"},[s("span",{class:"mtight"}," ")]),s("span",{class:"mord mathrm mtight"},"d")]),s("span",{class:"mord mathnormal mtight"},"λ")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathrm mtight"},"d"),s("span",{class:"mord mathnormal mtight"},"n")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])])])],-1),b=s("h3",{id:"_7-7-角度距離",tabindex:"-1"},[a("7.7: 角度距離 "),s("a",{class:"header-anchor",href:"#_7-7-角度距離","aria-label":'Permalink to "7.7: 角度距離"'},"​")],-1),k=s("ol",{start:"7"},[s("li",null,[a("理想的な望遠鏡 (レンズ) で 2 点を解像するときの角度距離 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"≈"),s("mn",null,"1.22"),s("mi",null,"λ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"\\varphi \\approx 1.22 \\lambda / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6776em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1.22"),s("span",{class:"mord mathnormal"},"λ"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(". この角度では, 一方の点の中 心が他方の点の最初の回折最小值に当たる.")])],-1),_=s("h3",{id:"_7-8-bragg-の法則",tabindex:"-1"},[a("7.8: Bragg の法則 "),s("a",{class:"header-anchor",href:"#_7-8-bragg-の法則","aria-label":'Permalink to "7.8: Bragg の法則"'},"​")],-1),z=s("ol",{start:"8"},[s("li",null,[a("Bragg の法則：間隔が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" の平行な結晶面の組は, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"d"),s("mi",null,"sin"),s("mo",null,"⁡"),s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"n"),s("mi",null,"λ")]),s("annotation",{encoding:"application/x-tex"},"2 d \\sin \\theta=n \\lambda")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"sin"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"nλ")])])]),a(" ならば "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"X")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{X}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathrm"},"X")])])]),a(" 線を反射する. ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"θ")]),s("annotation",{encoding:"application/x-tex"},"\\theta")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"θ")])])]),a(" は結 晶面と X 線がなす角 (かすめ角).")])],-1),f=s("h3",{id:"_7-9-高密度電体媒質反射",tabindex:"-1"},[a("7.9: 高密度電体媒質反射 "),s("a",{class:"header-anchor",href:"#_7-9-高密度電体媒質反射","aria-label":'Permalink to "7.9: 高密度電体媒質反射"'},"​")],-1),M=s("ol",{start:"9"},[s("li",null,[a("光学的に高密度な誘電体媒質による反射 : 位相が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(" ず れる. 半透明の薄膜では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")]),s("mo",null,"+"),s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")]),s("mo",null,"="),s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}+\\phi_{\\leftarrow}=\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),a(". ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"→")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\rightarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"→")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"ϕ"),s("mo",{lspace:"0em",rspace:"0em"},"←")])]),s("annotation",{encoding:"application/x-tex"},"\\phi_{\\leftarrow}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1068em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"←")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),a(" は反射波と透過波の位相差（矢印は入射方向を 示す)")])],-1),L=s("h3",{id:"_7-10-fabry-perot-干渉計",tabindex:"-1"},[a("7.10: Fabry-Pérot 干渉計 "),s("a",{class:"header-anchor",href:"#_7-10-fabry-perot-干渉計","aria-label":'Permalink to "7.10: Fabry-Pérot 干渉計"'},"​")],-1),P=s("ol",{start:"10"},[s("li",null,[a("Fabry-Pérot 干渉計 : 高い反射率 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",null,"≪"),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"r(1-r \\ll 1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≪"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" を持 つ 2 枚の平行な半透明の鏡. 分解能は "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("mi",null,"ν"),s("mrow",null,[s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"ν")])]),s("mo",null,"≈"),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"a")]),s("mrow",null,[s("mi",null,"λ"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"−"),s("mi",null,"r"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\frac{\\nu}{\\Delta \\nu} \\approx \\frac{2 a}{\\lambda(1-r)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"Δ"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.06366em"}},"ν")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.3651em","vertical-align":"-0.52em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8451em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"λ"),s("span",{class:"mopen mtight"},"("),s("span",{class:"mord mtight"},"1"),s("span",{class:"mbin mtight"},"−"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose mtight"},")")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight"},"a")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.52em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 5 つの平面波 (干渉計の前で左右に進む波, 内部を左右 に進む波，後ろを進む波）を設定して境界条件を課す ことで，透過スペクトルを求められる.")])],-1),B=s("h3",{id:"_7-11-コヒーレントな電磁波",tabindex:"-1"},[a("7.11: コヒーレントな電磁波 "),s("a",{class:"header-anchor",href:"#_7-11-コヒーレントな電磁波","aria-label":'Permalink to "7.11: コヒーレントな電磁波"'},"​")],-1),I=s("ol",{start:"11"},[s("li",null,[a("コヒーレントな電磁波: 電場をベクトル为で表し, ベク トル間の角度を位相差とする. 屈折率が "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"="),s("mi",null,"n"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"n=n(\\omega)=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mi",null,"ε"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mo",{stretchy:"false"},")")])])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{\\varepsilon(\\omega)}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.24em","vertical-align":"-0.305em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.935em"}},[s("span",{class:"svg-align",style:{top:"-3.2em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"mord",style:{"padding-left":"1em"}},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"mclose"},")")])]),s("span",{style:{top:"-2.895em"}},[s("span",{class:"pstrut",style:{height:"3.2em"}}),s("span",{class:"hide-tail",style:{"min-width":"1.02em",height:"1.28em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.28em",viewBox:"0 0 400000 1296",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 diff --git a/assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.js b/assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.js rename to assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.js index 0776138b..07a8ebb7 100644 --- a/assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.js +++ b/assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const N=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 8","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/8.md","filePath":"academic/physics/ipho-formulas-jpn/8.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/8.md"},i=l('

Formulas for IPhO 日本語版: Section 8 ​

8: 電気回路 ​

8.1: V=I R, P=V I ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"V"),s("mo",null,"="),s("mi",null,"I"),s("mi",null,"R"),s("mo",{separator:"true"},","),s("mi",null,"P"),s("mo",null,"="),s("mi",null,"V"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"V=I R, P=V I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"R"),s("mtext",null,"直列 ")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"R"),s("mtext",null,"並列 "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"R"),s("mi",null,"i"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"R_{\\text {直列 }}=\\sum R_i, R_{\\text {並列 }}^{-1}=\\sum R_i^{-1} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"直列"),s("span",{class:"mord mtight"}," ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.4163em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"並列"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2837em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.433em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.267em"}},[s("span")])])])])])])])])])])])],-1),p=s("h3",{id:"_8-2-kirchhoff-の法則",tabindex:"-1"},[a("8.2: Kirchhoff の法則 "),s("a",{class:"header-anchor",href:"#_8-2-kirchhoff-の法則","aria-label":'Permalink to "8.2: Kirchhoff の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[a("Kirchhoff の法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("munder",null,[s("mo",null,"∑"),s("mstyle",{scriptlevel:"1"},[s("mtable",{rowspacing:"0.1em",columnalign:"center",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"1",displaystyle:"false"},[s("mtext",null," 節点 ")])])])])])]),s("mi",null,"I"),s("mo",null,"="),s("mn",null,"0"),s("mo",{separator:"true"},","),s("munder",null,[s("mo",null,"∑"),s("mtext",null,"閉路 ")]),s("mi",null,"V"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\sum_{\\substack{\\text { 節点 }}} I=0, \\sum_{\\text {閉路 }} V=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4703em","vertical-align":"-1.4203em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8568em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6817em"}},[s("span",{style:{top:"-2.6983em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," "),s("span",{class:"mord cjk_fallback mtight"},"節点"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1817em"}},[s("span")])])])])])])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4203em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.3443em","vertical-align":"-1.2943em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8557em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"閉路"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2943em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),h=s("h3",{id:"_8-3-ポイント-2-の方程式を減らすために",tabindex:"-1"},[a("8.3: ポイント 2 の方程式を減らすために "),s("a",{class:"header-anchor",href:"#_8-3-ポイント-2-の方程式を減らすために","aria-label":'Permalink to "8.3: ポイント 2 の方程式を減らすために"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[a("ポイント 2 の方程式を減らすために: 節点電位法. ルー プ電流法. 等価回路 (3 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒"),s("mi",{mathvariant:"normal"},"△"),s("mtext",null,"又")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow \\triangle 又")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"△"),s("span",{class:"mord cjk_fallback"},"又")])])]),a(" 又 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Y")]),s("annotation",{encoding:"application/x-tex"},"Y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"Y")])])]),a(" の形, 起電力のある 2 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒")])])]),a(" 抵抗と電池の直列)")])],-1),g=l('

8.4: 無限につながる抵抗 ​

  1. 無限につながる抵抗 : 無限に続く格子の隣り合う節点 間で，自己相似性を使う。鏡像法の一般化された方法.

8.5: 交流回路 ​

',3),u=s("ol",{start:"5"},[s("li",null,[a("交流回路: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(" を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Z")]),s("annotation",{encoding:"application/x-tex"},"Z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z")])])]),a(" に置き換えてポイント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"∼"),s("mn",null,"4")]),s("annotation",{encoding:"application/x-tex"},"1 \\sim 4")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∼"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4")])])]),a(" を用 いる."),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msub",null,[s("mi",null,"Z"),s("mi",null,"R")]),s("mo",null,"="),s("mi",null,"R"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"C")]),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"L")]),s("mo",null,"="),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"L")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"V"),s("mtext",null,"eff ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"Z"),s("mi",{mathvariant:"normal"},"∣"),s("msub",null,[s("mi",null,"I"),s("mtext",null,"eff ")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"P"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"I"),s("mi",null,"i"),s("mn",null,"2")]),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} Z_R=R, Z_C=1 / i \\omega C, Z_L=i \\omega L \\\\ \\varphi=\\arg Z, V_{\\text {eff }}=|Z| I_{\\text {eff }} \\\\ P=|V||I| \\cos (\\arg Z)=\\sum I_i^2 R_i \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.9em","vertical-align":"-2.2em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.7em"}},[s("span",{style:{top:"-4.91em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal"},"L")])]),s("span",{style:{top:"-3.41em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.2222em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.7em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"∣∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.2em"}},[s("span")])])])])])])])])])])])])],-1),d=s("h3",{id:"_8-6-特性時間",tabindex:"-1"},[a("8.6: 特性時間 "),s("a",{class:"header-anchor",href:"#_8-6-特性時間","aria-label":'Permalink to "8.6: 特性時間"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("特性時間: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"R"),s("mi",null,"C")])]),s("mo",null,"="),s("mi",null,"R"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"R")])]),s("mo",null,"="),s("mi",null,"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"."),s("msub",null,[s("mi",null,"ω"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\tau_{R C}=R C, \\tau_{L R}=L / R . \\omega_{L C}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"RC")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"RC"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"."),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])])]),s("annotation",{encoding:"application/x-tex"},"1 / \\sqrt{L C}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1767em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9267em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.8867em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 +import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const N=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 8","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/8.md","filePath":"academic/physics/ipho-formulas-jpn/8.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/8.md"},i=l('

Formulas for IPhO 日本語版: Section 8 ​

8: 電気回路 ​

8.1: V=I R, P=V I ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"V"),s("mo",null,"="),s("mi",null,"I"),s("mi",null,"R"),s("mo",{separator:"true"},","),s("mi",null,"P"),s("mo",null,"="),s("mi",null,"V"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"V=I R, P=V I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"R"),s("mtext",null,"直列 ")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"R"),s("mtext",null,"並列 "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"R"),s("mi",null,"i"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"R_{\\text {直列 }}=\\sum R_i, R_{\\text {並列 }}^{-1}=\\sum R_i^{-1} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"直列"),s("span",{class:"mord mtight"}," ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.4163em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"並列"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2837em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.433em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.267em"}},[s("span")])])])])])])])])])])])],-1),p=s("h3",{id:"_8-2-kirchhoff-の法則",tabindex:"-1"},[a("8.2: Kirchhoff の法則 "),s("a",{class:"header-anchor",href:"#_8-2-kirchhoff-の法則","aria-label":'Permalink to "8.2: Kirchhoff の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[a("Kirchhoff の法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("munder",null,[s("mo",null,"∑"),s("mstyle",{scriptlevel:"1"},[s("mtable",{rowspacing:"0.1em",columnalign:"center",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"1",displaystyle:"false"},[s("mtext",null," 節点 ")])])])])])]),s("mi",null,"I"),s("mo",null,"="),s("mn",null,"0"),s("mo",{separator:"true"},","),s("munder",null,[s("mo",null,"∑"),s("mtext",null,"閉路 ")]),s("mi",null,"V"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\sum_{\\substack{\\text { 節点 }}} I=0, \\sum_{\\text {閉路 }} V=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4703em","vertical-align":"-1.4203em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8568em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6817em"}},[s("span",{style:{top:"-2.6983em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," "),s("span",{class:"mord cjk_fallback mtight"},"節点"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1817em"}},[s("span")])])])])])])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4203em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.3443em","vertical-align":"-1.2943em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8557em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"閉路"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2943em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),h=s("h3",{id:"_8-3-ポイント-2-の方程式を減らすために",tabindex:"-1"},[a("8.3: ポイント 2 の方程式を減らすために "),s("a",{class:"header-anchor",href:"#_8-3-ポイント-2-の方程式を減らすために","aria-label":'Permalink to "8.3: ポイント 2 の方程式を減らすために"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[a("ポイント 2 の方程式を減らすために: 節点電位法. ルー プ電流法. 等価回路 (3 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒"),s("mi",{mathvariant:"normal"},"△"),s("mtext",null,"又")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow \\triangle 又")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"△"),s("span",{class:"mord cjk_fallback"},"又")])])]),a(" 又 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Y")]),s("annotation",{encoding:"application/x-tex"},"Y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"Y")])])]),a(" の形, 起電力のある 2 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒")])])]),a(" 抵抗と電池の直列)")])],-1),g=l('

8.4: 無限につながる抵抗 ​

  1. 無限につながる抵抗 : 無限に続く格子の隣り合う節点 間で，自己相似性を使う。鏡像法の一般化された方法.

8.5: 交流回路 ​

',3),u=s("ol",{start:"5"},[s("li",null,[a("交流回路: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(" を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Z")]),s("annotation",{encoding:"application/x-tex"},"Z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z")])])]),a(" に置き換えてポイント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"∼"),s("mn",null,"4")]),s("annotation",{encoding:"application/x-tex"},"1 \\sim 4")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∼"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4")])])]),a(" を用 いる."),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msub",null,[s("mi",null,"Z"),s("mi",null,"R")]),s("mo",null,"="),s("mi",null,"R"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"C")]),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"L")]),s("mo",null,"="),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"L")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"V"),s("mtext",null,"eff ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"Z"),s("mi",{mathvariant:"normal"},"∣"),s("msub",null,[s("mi",null,"I"),s("mtext",null,"eff ")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"P"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"I"),s("mi",null,"i"),s("mn",null,"2")]),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} Z_R=R, Z_C=1 / i \\omega C, Z_L=i \\omega L \\\\ \\varphi=\\arg Z, V_{\\text {eff }}=|Z| I_{\\text {eff }} \\\\ P=|V||I| \\cos (\\arg Z)=\\sum I_i^2 R_i \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.9em","vertical-align":"-2.2em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.7em"}},[s("span",{style:{top:"-4.91em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal"},"L")])]),s("span",{style:{top:"-3.41em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.2222em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.7em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"∣∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.2em"}},[s("span")])])])])])])])])])])])])],-1),d=s("h3",{id:"_8-6-特性時間",tabindex:"-1"},[a("8.6: 特性時間 "),s("a",{class:"header-anchor",href:"#_8-6-特性時間","aria-label":'Permalink to "8.6: 特性時間"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("特性時間: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"R"),s("mi",null,"C")])]),s("mo",null,"="),s("mi",null,"R"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"R")])]),s("mo",null,"="),s("mi",null,"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"."),s("msub",null,[s("mi",null,"ω"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\tau_{R C}=R C, \\tau_{L R}=L / R . \\omega_{L C}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"RC")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"RC"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"."),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])])]),s("annotation",{encoding:"application/x-tex"},"1 / \\sqrt{L C}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1767em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9267em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.8867em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 diff --git a/assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.lean.js b/assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.lean.js rename to assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.lean.js index 40f25c76..7691dd1e 100644 --- a/assets/academic_physics_ipho-formulas-jpn_8.md.544b0b43.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_8.md.02aeb40e.lean.js @@ -1,4 +1,4 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const N=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 8","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/8.md","filePath":"academic/physics/ipho-formulas-jpn/8.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/8.md"},i=l("",3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"V"),s("mo",null,"="),s("mi",null,"I"),s("mi",null,"R"),s("mo",{separator:"true"},","),s("mi",null,"P"),s("mo",null,"="),s("mi",null,"V"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"V=I R, P=V I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"R"),s("mtext",null,"直列 ")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"R"),s("mtext",null,"並列 "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"R"),s("mi",null,"i"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"R_{\\text {直列 }}=\\sum R_i, R_{\\text {並列 }}^{-1}=\\sum R_i^{-1} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"直列"),s("span",{class:"mord mtight"}," ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.4163em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"並列"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2837em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.433em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.267em"}},[s("span")])])])])])])])])])])])],-1),p=s("h3",{id:"_8-2-kirchhoff-の法則",tabindex:"-1"},[a("8.2: Kirchhoff の法則 "),s("a",{class:"header-anchor",href:"#_8-2-kirchhoff-の法則","aria-label":'Permalink to "8.2: Kirchhoff の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[a("Kirchhoff の法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("munder",null,[s("mo",null,"∑"),s("mstyle",{scriptlevel:"1"},[s("mtable",{rowspacing:"0.1em",columnalign:"center",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"1",displaystyle:"false"},[s("mtext",null," 節点 ")])])])])])]),s("mi",null,"I"),s("mo",null,"="),s("mn",null,"0"),s("mo",{separator:"true"},","),s("munder",null,[s("mo",null,"∑"),s("mtext",null,"閉路 ")]),s("mi",null,"V"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\sum_{\\substack{\\text { 節点 }}} I=0, \\sum_{\\text {閉路 }} V=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4703em","vertical-align":"-1.4203em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8568em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6817em"}},[s("span",{style:{top:"-2.6983em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," "),s("span",{class:"mord cjk_fallback mtight"},"節点"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1817em"}},[s("span")])])])])])])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4203em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.3443em","vertical-align":"-1.2943em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8557em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"閉路"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2943em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),h=s("h3",{id:"_8-3-ポイント-2-の方程式を減らすために",tabindex:"-1"},[a("8.3: ポイント 2 の方程式を減らすために "),s("a",{class:"header-anchor",href:"#_8-3-ポイント-2-の方程式を減らすために","aria-label":'Permalink to "8.3: ポイント 2 の方程式を減らすために"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[a("ポイント 2 の方程式を減らすために: 節点電位法. ルー プ電流法. 等価回路 (3 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒"),s("mi",{mathvariant:"normal"},"△"),s("mtext",null,"又")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow \\triangle 又")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"△"),s("span",{class:"mord cjk_fallback"},"又")])])]),a(" 又 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Y")]),s("annotation",{encoding:"application/x-tex"},"Y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"Y")])])]),a(" の形, 起電力のある 2 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒")])])]),a(" 抵抗と電池の直列)")])],-1),g=l("",3),u=s("ol",{start:"5"},[s("li",null,[a("交流回路: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(" を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Z")]),s("annotation",{encoding:"application/x-tex"},"Z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z")])])]),a(" に置き換えてポイント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"∼"),s("mn",null,"4")]),s("annotation",{encoding:"application/x-tex"},"1 \\sim 4")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∼"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4")])])]),a(" を用 いる."),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msub",null,[s("mi",null,"Z"),s("mi",null,"R")]),s("mo",null,"="),s("mi",null,"R"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"C")]),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"L")]),s("mo",null,"="),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"L")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"V"),s("mtext",null,"eff ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"Z"),s("mi",{mathvariant:"normal"},"∣"),s("msub",null,[s("mi",null,"I"),s("mtext",null,"eff ")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"P"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"I"),s("mi",null,"i"),s("mn",null,"2")]),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} Z_R=R, Z_C=1 / i \\omega C, Z_L=i \\omega L \\\\ \\varphi=\\arg Z, V_{\\text {eff }}=|Z| I_{\\text {eff }} \\\\ P=|V||I| \\cos (\\arg Z)=\\sum I_i^2 R_i \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.9em","vertical-align":"-2.2em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.7em"}},[s("span",{style:{top:"-4.91em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal"},"L")])]),s("span",{style:{top:"-3.41em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.2222em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.7em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"∣∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.2em"}},[s("span")])])])])])])])])])])])])],-1),d=s("h3",{id:"_8-6-特性時間",tabindex:"-1"},[a("8.6: 特性時間 "),s("a",{class:"header-anchor",href:"#_8-6-特性時間","aria-label":'Permalink to "8.6: 特性時間"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("特性時間: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"R"),s("mi",null,"C")])]),s("mo",null,"="),s("mi",null,"R"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"R")])]),s("mo",null,"="),s("mi",null,"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"."),s("msub",null,[s("mi",null,"ω"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\tau_{R C}=R C, \\tau_{L R}=L / R . \\omega_{L C}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"RC")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"RC"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"."),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])])]),s("annotation",{encoding:"application/x-tex"},"1 / \\sqrt{L C}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1767em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9267em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.8867em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 +import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const N=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 8","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/8.md","filePath":"academic/physics/ipho-formulas-jpn/8.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/8.md"},i=l("",3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"V"),s("mo",null,"="),s("mi",null,"I"),s("mi",null,"R"),s("mo",{separator:"true"},","),s("mi",null,"P"),s("mo",null,"="),s("mi",null,"V"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"V=I R, P=V I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"R"),s("mtext",null,"直列 ")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")]),s("mo",{separator:"true"},","),s("msubsup",null,[s("mi",null,"R"),s("mtext",null,"並列 "),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"R"),s("mi",null,"i"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"R_{\\text {直列 }}=\\sum R_i, R_{\\text {並列 }}^{-1}=\\sum R_i^{-1} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8333em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"直列"),s("span",{class:"mord mtight"}," ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.4163em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"並列"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2837em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.433em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.267em"}},[s("span")])])])])])])])])])])])],-1),p=s("h3",{id:"_8-2-kirchhoff-の法則",tabindex:"-1"},[a("8.2: Kirchhoff の法則 "),s("a",{class:"header-anchor",href:"#_8-2-kirchhoff-の法則","aria-label":'Permalink to "8.2: Kirchhoff の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[a("Kirchhoff の法則 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("munder",null,[s("mo",null,"∑"),s("mstyle",{scriptlevel:"1"},[s("mtable",{rowspacing:"0.1em",columnalign:"center",columnspacing:"1em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"1",displaystyle:"false"},[s("mtext",null," 節点 ")])])])])])]),s("mi",null,"I"),s("mo",null,"="),s("mn",null,"0"),s("mo",{separator:"true"},","),s("munder",null,[s("mo",null,"∑"),s("mtext",null,"閉路 ")]),s("mi",null,"V"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\sum_{\\substack{\\text { 節点 }}} I=0, \\sum_{\\text {閉路 }} V=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4703em","vertical-align":"-1.4203em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8568em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6817em"}},[s("span",{style:{top:"-2.6983em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"}," "),s("span",{class:"mord cjk_fallback mtight"},"節点"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1817em"}},[s("span")])])])])])])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.4203em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.3443em","vertical-align":"-1.2943em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.05em"}},[s("span",{style:{top:"-1.8557em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord cjk_fallback mtight"},"閉路"),s("span",{class:"mord mtight"}," ")])])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2943em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),h=s("h3",{id:"_8-3-ポイント-2-の方程式を減らすために",tabindex:"-1"},[a("8.3: ポイント 2 の方程式を減らすために "),s("a",{class:"header-anchor",href:"#_8-3-ポイント-2-の方程式を減らすために","aria-label":'Permalink to "8.3: ポイント 2 の方程式を減らすために"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[a("ポイント 2 の方程式を減らすために: 節点電位法. ルー プ電流法. 等価回路 (3 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒"),s("mi",{mathvariant:"normal"},"△"),s("mtext",null,"又")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow \\triangle 又")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"△"),s("span",{class:"mord cjk_fallback"},"又")])])]),a(" 又 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Y")]),s("annotation",{encoding:"application/x-tex"},"Y")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"Y")])])]),a(" の形, 起電力のある 2 端子の場合 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"⇒")]),s("annotation",{encoding:"application/x-tex"},"\\Rightarrow")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.3669em"}}),s("span",{class:"mrel"},"⇒")])])]),a(" 抵抗と電池の直列)")])],-1),g=l("",3),u=s("ol",{start:"5"},[s("li",null,[a("交流回路: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(" を "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Z")]),s("annotation",{encoding:"application/x-tex"},"Z")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z")])])]),a(" に置き換えてポイント "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mo",null,"∼"),s("mn",null,"4")]),s("annotation",{encoding:"application/x-tex"},"1 \\sim 4")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∼"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4")])])]),a(" を用 いる."),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"center",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("msub",null,[s("mi",null,"Z"),s("mi",null,"R")]),s("mo",null,"="),s("mi",null,"R"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"C")]),s("mo",null,"="),s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"Z"),s("mi",null,"L")]),s("mo",null,"="),s("mi",null,"i"),s("mi",null,"ω"),s("mi",null,"L")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"φ"),s("mo",null,"="),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"V"),s("mtext",null,"eff ")]),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"Z"),s("mi",{mathvariant:"normal"},"∣"),s("msub",null,[s("mi",null,"I"),s("mtext",null,"eff ")])])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mi",null,"P"),s("mo",null,"="),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"V"),s("mi",{mathvariant:"normal"},"∣"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"∣"),s("mi",null,"cos"),s("mo",null,"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"arg"),s("mo",null,"⁡"),s("mi",null,"Z"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mo",null,"∑"),s("msubsup",null,[s("mi",null,"I"),s("mi",null,"i"),s("mn",null,"2")]),s("msub",null,[s("mi",null,"R"),s("mi",null,"i")])])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{gathered} Z_R=R, Z_C=1 / i \\omega C, Z_L=i \\omega L \\\\ \\varphi=\\arg Z, V_{\\text {eff }}=|Z| I_{\\text {eff }} \\\\ P=|V||I| \\cos (\\arg Z)=\\sum I_i^2 R_i \\end{gathered} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"4.9em","vertical-align":"-2.2em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-c"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.7em"}},[s("span",{style:{top:"-4.91em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0715em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"L")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"iω"),s("span",{class:"mord mathnormal"},"L")])]),s("span",{style:{top:"-3.41em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.2222em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mord"},"∣"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3361em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord text mtight"},[s("span",{class:"mord mtight"},"eff ")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])]),s("span",{style:{top:"-1.7em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"P"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},"∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V"),s("span",{class:"mord"},"∣∣"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},"∣"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop"},"cos"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("ar"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"Z"),s("span",{class:"mclose"},")"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8641em"}},[s("span",{style:{top:"-2.453em","margin-left":"-0.0785em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])]),s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.247em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0077em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"2.2em"}},[s("span")])])])])])])])])])])])])],-1),d=s("h3",{id:"_8-6-特性時間",tabindex:"-1"},[a("8.6: 特性時間 "),s("a",{class:"header-anchor",href:"#_8-6-特性時間","aria-label":'Permalink to "8.6: 特性時間"'},"​")],-1),y=s("ol",{start:"6"},[s("li",null,[a("特性時間: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"R"),s("mi",null,"C")])]),s("mo",null,"="),s("mi",null,"R"),s("mi",null,"C"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",null,"τ"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"R")])]),s("mo",null,"="),s("mi",null,"L"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"."),s("msub",null,[s("mi",null,"ω"),s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])]),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"\\tau_{R C}=R C, \\tau_{L R}=L / R . \\omega_{L C}=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"RC")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"RC"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.1132em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"."),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3283em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"L"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07153em"}},"C")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("mi",{mathvariant:"normal"},"/"),s("msqrt",null,[s("mrow",null,[s("mi",null,"L"),s("mi",null,"C")])])]),s("annotation",{encoding:"application/x-tex"},"1 / \\sqrt{L C}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1767em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1/"),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9267em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C")])]),s("span",{style:{top:"-2.8867em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 diff --git a/assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.js b/assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.js rename to assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.js index 95413857..f8d118aa 100644 --- a/assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.js +++ b/assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.js @@ -1 +1 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const Y=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 9","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/9.md","filePath":"academic/physics/ipho-formulas-jpn/9.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/9.md"},i=l('

Formulas for IPhO 日本語版: Section 9 ​

9: 電磁気学 ​

9.1: Coulomb の法則 ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"F=k q_1 q_2 / r^2, U=k q_1 q_2 / r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" で, Kepler の法則が 使える ("),s("a",{href:"./12"},"Section 12 参照"),a(").")])],-1),p=s("h3",{id:"_9-2-gauss-の法則",tabindex:"-1"},[a("9.2: Gauss の法則 "),s("a",{class:"header-anchor",href:"#_9-2-gauss-の法則","aria-label":'Permalink to "9.2: Gauss の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[s("p",null,[a("Gauss の法則 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{S}=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.3061em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(",")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",null,"ε"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mi",null,"Q"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mo",null,"−"),s("mn",null,"4"),s("mi",null,"π"),s("mi",null,"G"),s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\varepsilon \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{S}=Q, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{S}=-4 \\pi G M ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal"},"Q"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM")])])])])])])],-1),h=s("h3",{id:"_9-3-循環定理",tabindex:"-1"},[a("9.3: 循環定理 "),s("a",{class:"header-anchor",href:"#_9-3-循環定理","aria-label":'Permalink to "9.3: 循環定理"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[s("p",null,"循環定理 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mo",null,"="),s("mover",{accent:"true"},[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"˙")]),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l")]),s("mi",null,"μ")]),s("mo",null,"="),s("mi",null,"I"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{l}=0(=\\dot{\\Phi}), \\oint \\frac{\\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{l}}{\\mu}=I, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{l}=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2519em","vertical-align":"-0.8804em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9202em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},"Φ")]),s("span",{style:{top:"-3.2523em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),g=s("h3",{id:"_9-4-電流素片により生じる磁束密度",tabindex:"-1"},[a("9.4: 電流素片により生じる磁束密度 "),s("a",{class:"header-anchor",href:"#_9-4-電流素片により生じる磁束密度","aria-label":'Permalink to "9.4: 電流素片により生じる磁束密度"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("電流素片により生じる磁束密度 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")])]),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{d} \\boldsymbol{B}=\\frac{\\mu I}{4 \\pi} \\frac{\\mathrm{d} \\boldsymbol{l} \\times \\boldsymbol{e}_r}{r^2} . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0574em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3603em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},".")])])])])]),a(" したがって電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(" が流れる円形回路の中心では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),u=s("h3",{id:"_9-5-ローレンツ力",tabindex:"-1"},[a("9.5: ローレンツ力 "),s("a",{class:"header-anchor",href:"#_9-5-ローレンツ力","aria-label":'Permalink to "9.5: ローレンツ力"'},"​")],-1),y=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"e"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"I"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=e(\\boldsymbol{E}+\\boldsymbol{v} \\times \\boldsymbol{B}), \\boldsymbol{F}=\\boldsymbol{I} \\times \\boldsymbol{B} l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"e"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7694em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.07778em"}},"I")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(".")])],-1),b=s("h3",{id:"_9-6-gauss-の定理と循環定理より",tabindex:"-1"},[a("9.6: Gauss の定理と循環定理より "),s("a",{class:"header-anchor",href:"#_9-6-gauss-の定理と循環定理より","aria-label":'Permalink to "9.6: Gauss の定理と循環定理より"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,[a("Gauss の定理と循環定理より：帯電した導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")]),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\pi \\varepsilon_0 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 \\pi r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 帯電した面について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")])])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\varepsilon_0}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる面につい て "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"i")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 i}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2528em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9078em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 一様に帯電した球殼（又は無限に長い円 筒）の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"E=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(", 軸に沿って表面に電流が流れる 円筒の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" で一様に帯電, 又は一様 な電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"i")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{i}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6933em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])])])])]),a(" が流れる, 球 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"3"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d=3) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"3"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 円柱 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"−"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d-2) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 平面 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(d=1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" の内部で,"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"ρ"),s("mrow",null,[s("mi",null,"ε"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mrow",null,[s("mi",null,"μ"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"i"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"r")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{E}=\\frac{\\rho}{\\varepsilon d} \\boldsymbol{r}, \\boldsymbol{B}=\\frac{1}{\\mu d} \\boldsymbol{i} \\times \\boldsymbol{r} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.7936em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2019em","vertical-align":"-0.8804em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])])])])])])])])],-1),x=s("h3",{id:"_9-7-長いソレノイド",tabindex:"-1"},[a("9.7: 長いソレノイド "),s("a",{class:"header-anchor",href:"#_9-7-長いソレノイド","aria-label":'Permalink to "9.7: 長いソレノイド"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("長いソレノイド: 内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"B=\\mu n I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(", 外部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 磁束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"B"),s("mi",null,"S"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"N"),s("mi",null,"l")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=N B S\\left(n=\\frac{N}{l}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2223em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NBS"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.01968em"}},"l")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"N")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])]),a(". インダクタンス "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"L=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"I"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"\\Phi / I=\\mu n^2 V")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V")])])]),a(". 短いソレノイド "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"B"),s("mi",{mathvariant:"normal"},"∥")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"Ω")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},": B_{\\|}=\\frac{\\mu n I \\Omega}{4 \\pi}(\\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0385em","vertical-align":"-0.3552em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.5198em","margin-left":"-0.0502em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"∥")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3552em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mtight"},"Ω")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Ω")])])]),a(" は 立体角).")])],-1),k=s("h3",{id:"_9-8-磁場を小型コイルや衝撃検流計で測定する",tabindex:"-1"},[a("9.8: 磁場を小型コイルや衝撃検流計で測定する "),s("a",{class:"header-anchor",href:"#_9-8-磁場を小型コイルや衝撃検流計で測定する","aria-label":'Permalink to "9.8: 磁場を小型コイルや衝撃検流計で測定する"'},"​")],-1),_=s("ol",{start:"9"},[s("li",null,[a("磁場を小型コイルや衝撃検流計で測定する： "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"q"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"q=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∫"),s("mfrac",null,[s("mi",null,"V"),s("mi",null,"R")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"S"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"B"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"\\int \\frac{V}{R} \\mathrm{~d} t=N S \\Delta B / R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.22222em"}},"V")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NS"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(".")])],-1),z=s("h3",{id:"_9-9-静電場のエネルギー",tabindex:"-1"},[a("9.9: 静電場のエネルギー "),s("a",{class:"header-anchor",href:"#_9-9-静電場のエネルギー","aria-label":'Permalink to "9.9: 静電場のエネルギー"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("静電場のエネルギー："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("munder",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"<"),s("mi",null,"j")])]),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"q"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"q"),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"r"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"j")])])]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",null,"∫"),s("mi",null,"ϕ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"q"),s("mo",{separator:"true"},","),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"q"),s("mo",null,"="),s("mi",null,"ρ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"U=k \\sum_{i9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​
  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

',3),B=s("ol",{start:"12"},[s("li",null,[a("全ての電荷が距離 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" にある場合（例えば，不均一に帯 電した球やリングの中心） "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ϕ"),s("mi",null,"ϕ"),s("mo",null,"="),s("mi",null,"k"),s("mi",null,"Q"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"\\phi \\phi=k Q / r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ϕϕ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord mathnormal"},"Q"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])])])],-1),q=l('

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

',3),L=s("ol",{start:"14"},[s("li",null,[a("導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Q")]),s("annotation",{encoding:"application/x-tex"},"Q")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"Q")])])]),a(" という 電荷を持った導電性の球があるように見える).")])],-1),I=s("h3",{id:"_9-15-静電容量",tabindex:"-1"},[a("9.15: 静電容量 "),s("a",{class:"header-anchor",href:"#_9-15-静電容量","aria-label":'Permalink to "9.15: 静電容量"'},"​")],-1),P=s("ol",{start:"15"},[s("li",null,[a("静電容量: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mo",null,"="),s("mi",null,"ε"),s("mi",null,"S"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"C=\\varepsilon S / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"εS"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(" (平板), "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"4"),s("mi",null,"π"),s("mi",null,"ε"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"4 \\pi \\varepsilon r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" (球), "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"ε"),s("mi",null,"l"),s("mo",{stretchy:"false"},"("),s("mi",null,"ln"),s("mo",null,"⁡"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"2 \\pi \\varepsilon l(\\ln R / r)^{-1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"εl"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"ln"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),a(" (同軸円筒).")])],-1),S=s("h3",{id:"_9-16-双極子モーメント",tabindex:"-1"},[a("9.16: 双極子モーメント "),s("a",{class:"header-anchor",href:"#_9-16-双極子モーメント","aria-label":'Permalink to "9.16: 双極子モーメント"'},"​")],-1),E=s("ol",{start:"16"},[s("li",null,[s("p",null,"双極子モーメント："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"e")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"q"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"="),s("mi",null,"q"),s("mi",{mathvariant:"bold-italic"},"d"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"μ")]),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"S")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}_e=\\sum q_i \\boldsymbol{r}_i=q \\boldsymbol{d}, \\boldsymbol{p}_\\mu=I \\boldsymbol{S} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6886em","vertical-align":"-0.2441em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2441em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0747em","vertical-align":"-0.3802em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"d")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3802em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])])])])])])])])],-1),T=s("h3",{id:"_9-17-双極子場",tabindex:"-1"},[a("9.17: 双極子場 "),s("a",{class:"header-anchor",href:"#_9-17-双極子場","aria-label":'Permalink to "9.17: 双極子場"'},"​")],-1),F=s("ol",{start:"17"},[s("li",null,[a("双極子場 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ϕ"),s("mo",null,"="),s("mi",null,"k"),s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"⋅"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",{separator:"true"},","),s("mi",null,"B"),s("mo",null,"∝"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])])]),s("annotation",{encoding:"application/x-tex"},"\\phi=k \\boldsymbol{p} \\cdot \\boldsymbol{e}_r / r^2, E, B \\propto r^{-3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])])])])])])],-1),V=s("h3",{id:"_9-18-双極子に働く力",tabindex:"-1"},[a("9.18: 双極子に働く力 "),s("a",{class:"header-anchor",href:"#_9-18-双極子に働く力","aria-label":'Permalink to "9.18: 双極子に働く力"'},"​")],-1),N=s("ol",{start:"19"},[s("li",null,[a("双極子に働く力 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"e")]),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",{fence:"true"},")")]),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",{separator:"true"},","),s("mi",null,"F"),s("mo",null,"="),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"μ")]),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",{fence:"true"},")")]),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"F=\\left(\\boldsymbol{p}_e \\cdot \\boldsymbol{E}\\right)^{\\prime}, F=\\left(\\boldsymbol{p}_\\mu \\cdot \\boldsymbol{B}\\right)^{\\prime}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1418em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2441em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8918em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.372em","vertical-align":"-0.3802em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3802em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9918em"}},[s("span",{style:{top:"-3.3029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(" [訳 者注 : ここの微分はむしろ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"grad"),s("mo",null,"⁡")]),s("annotation",{encoding:"application/x-tex"},"\\operatorname{grad}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"grad")])])])]),a(" である]. 2 つの双極 子間の相互作用 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mi",null,"F"),s("mo",null,"∝"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])])]),s("annotation",{encoding:"application/x-tex"},": F \\propto r^{-4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])])])])]),a(".")])],-1),R=s("h3",{id:"_9-19-磁気双極子としての点電荷",tabindex:"-1"},[a("9.19: 磁気双極子としての点電荷 "),s("a",{class:"header-anchor",href:"#_9-19-磁気双極子としての点電荷","aria-label":'Permalink to "9.19: 磁気双極子としての点電荷"'},"​")],-1),C=s("ol",{start:"19"},[s("li",null,[a("磁気双極子としての点電荷 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"μ")]),s("mo",null,"∝"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"∝"),s("msubsup",null,[s("mi",null,"v"),s("mo",{lspace:"0em",rspace:"0em"},"⊥"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"p_\\mu \\propto \\Phi \\propto v_{\\perp}^2 / B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0972em","vertical-align":"-0.2831em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4169em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"⊥")])])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2831em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" は断熱 不変量 ("),s("a",{href:"./4#_4-22-断熱不変量"},"Section 4: #22"),a(" 参照).")])],-1),j=l('

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

',3),A=s("ol",{start:"21"},[s("li",null,[a("一様（電）場中の球 (円柱) の分極 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"+"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"(+\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"+"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"-\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" に一 様に帯電した球 (円柱) の重ね合わせで， "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"∝"),s("mi",null,"E")]),s("annotation",{encoding:"application/x-tex"},"d \\propto E")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E")])])]),a(".")])],-1),G=s("h3",{id:"_9-22-渦電流",tabindex:"-1"},[a("9.22: 渦電流 "),s("a",{class:"header-anchor",href:"#_9-22-渦電流","aria-label":'Permalink to "9.22: 渦電流"'},"​")],-1),Q=s("ol",{start:"22"},[s("li",null,[a("渦電流: 電流損失密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"≈"),s("msup",null,[s("mi",null,"B"),s("mn",null,"2")]),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"ρ"),s("mi",{mathvariant:"normal"},"."),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\approx B^2 v^2 / \\rho .1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4831em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord"},".1")])])]),a(" 回の通過で与え られる運動量 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mi",null,"τ"),s("mo",null,"≈"),s("msup",null,[s("mi",null,"B"),s("mn",null,"2")]),s("msup",null,[s("mi",null,"a"),s("mn",null,"3")]),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"F \\tau \\approx B^2 a^3 d / \\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])])])])])]),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" (ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" は厚さ, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" は大きさ).")])],-1),U=[i,r,p,c,h,o,g,d,u,y,b,v,x,w,k,_,z,f,M,B,q,L,I,P,S,E,T,F,V,N,R,C,j,A,G,Q];function D(O,$,J,K,H,W){return m(),n("div",null,U)}const Z=t(e,[["render",D]]);export{Y as __pageData,Z as default}; +import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const Y=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 9","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/9.md","filePath":"academic/physics/ipho-formulas-jpn/9.md","lastUpdated":1684859551000}'),e={name:"academic/physics/ipho-formulas-jpn/9.md"},i=l('

Formulas for IPhO 日本語版: Section 9 ​

9: 電磁気学 ​

9.1: Coulomb の法則 ​

',3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"F=k q_1 q_2 / r^2, U=k q_1 q_2 / r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" で, Kepler の法則が 使える ("),s("a",{href:"./12"},"Section 12 参照"),a(").")])],-1),p=s("h3",{id:"_9-2-gauss-の法則",tabindex:"-1"},[a("9.2: Gauss の法則 "),s("a",{class:"header-anchor",href:"#_9-2-gauss-の法則","aria-label":'Permalink to "9.2: Gauss の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[s("p",null,[a("Gauss の法則 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{S}=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.3061em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(",")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",null,"ε"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mi",null,"Q"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mo",null,"−"),s("mn",null,"4"),s("mi",null,"π"),s("mi",null,"G"),s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\varepsilon \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{S}=Q, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{S}=-4 \\pi G M ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal"},"Q"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM")])])])])])])],-1),h=s("h3",{id:"_9-3-循環定理",tabindex:"-1"},[a("9.3: 循環定理 "),s("a",{class:"header-anchor",href:"#_9-3-循環定理","aria-label":'Permalink to "9.3: 循環定理"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[s("p",null,"循環定理 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mo",null,"="),s("mover",{accent:"true"},[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"˙")]),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l")]),s("mi",null,"μ")]),s("mo",null,"="),s("mi",null,"I"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{l}=0(=\\dot{\\Phi}), \\oint \\frac{\\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{l}}{\\mu}=I, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{l}=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2519em","vertical-align":"-0.8804em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9202em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},"Φ")]),s("span",{style:{top:"-3.2523em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),g=s("h3",{id:"_9-4-電流素片により生じる磁束密度",tabindex:"-1"},[a("9.4: 電流素片により生じる磁束密度 "),s("a",{class:"header-anchor",href:"#_9-4-電流素片により生じる磁束密度","aria-label":'Permalink to "9.4: 電流素片により生じる磁束密度"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("電流素片により生じる磁束密度 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")])]),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{d} \\boldsymbol{B}=\\frac{\\mu I}{4 \\pi} \\frac{\\mathrm{d} \\boldsymbol{l} \\times \\boldsymbol{e}_r}{r^2} . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0574em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3603em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},".")])])])])]),a(" したがって電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(" が流れる円形回路の中心では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),u=s("h3",{id:"_9-5-ローレンツ力",tabindex:"-1"},[a("9.5: ローレンツ力 "),s("a",{class:"header-anchor",href:"#_9-5-ローレンツ力","aria-label":'Permalink to "9.5: ローレンツ力"'},"​")],-1),y=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"e"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"I"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=e(\\boldsymbol{E}+\\boldsymbol{v} \\times \\boldsymbol{B}), \\boldsymbol{F}=\\boldsymbol{I} \\times \\boldsymbol{B} l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"e"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7694em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.07778em"}},"I")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(".")])],-1),b=s("h3",{id:"_9-6-gauss-の定理と循環定理より",tabindex:"-1"},[a("9.6: Gauss の定理と循環定理より "),s("a",{class:"header-anchor",href:"#_9-6-gauss-の定理と循環定理より","aria-label":'Permalink to "9.6: Gauss の定理と循環定理より"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,[a("Gauss の定理と循環定理より：帯電した導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")]),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\pi \\varepsilon_0 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 \\pi r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 帯電した面について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")])])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\varepsilon_0}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる面につい て "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"i")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 i}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2528em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9078em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 一様に帯電した球殼（又は無限に長い円 筒）の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"E=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(", 軸に沿って表面に電流が流れる 円筒の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" で一様に帯電, 又は一様 な電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"i")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{i}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6933em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])])])])]),a(" が流れる, 球 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"3"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d=3) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"3"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 円柱 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"−"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d-2) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 平面 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(d=1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" の内部で,"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"ρ"),s("mrow",null,[s("mi",null,"ε"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mrow",null,[s("mi",null,"μ"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"i"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"r")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{E}=\\frac{\\rho}{\\varepsilon d} \\boldsymbol{r}, \\boldsymbol{B}=\\frac{1}{\\mu d} \\boldsymbol{i} \\times \\boldsymbol{r} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.7936em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2019em","vertical-align":"-0.8804em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])])])])])])])])],-1),x=s("h3",{id:"_9-7-長いソレノイド",tabindex:"-1"},[a("9.7: 長いソレノイド "),s("a",{class:"header-anchor",href:"#_9-7-長いソレノイド","aria-label":'Permalink to "9.7: 長いソレノイド"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("長いソレノイド: 内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"B=\\mu n I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(", 外部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 磁束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"B"),s("mi",null,"S"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"N"),s("mi",null,"l")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=N B S\\left(n=\\frac{N}{l}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2223em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NBS"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.01968em"}},"l")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"N")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])]),a(". インダクタンス "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"L=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"I"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"\\Phi / I=\\mu n^2 V")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V")])])]),a(". 短いソレノイド "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"B"),s("mi",{mathvariant:"normal"},"∥")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"Ω")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},": B_{\\|}=\\frac{\\mu n I \\Omega}{4 \\pi}(\\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0385em","vertical-align":"-0.3552em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.5198em","margin-left":"-0.0502em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"∥")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3552em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mtight"},"Ω")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Ω")])])]),a(" は 立体角).")])],-1),k=s("h3",{id:"_9-8-磁場を小型コイルや衝撃検流計で測定する",tabindex:"-1"},[a("9.8: 磁場を小型コイルや衝撃検流計で測定する "),s("a",{class:"header-anchor",href:"#_9-8-磁場を小型コイルや衝撃検流計で測定する","aria-label":'Permalink to "9.8: 磁場を小型コイルや衝撃検流計で測定する"'},"​")],-1),_=s("ol",{start:"9"},[s("li",null,[a("磁場を小型コイルや衝撃検流計で測定する： "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"q"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"q=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∫"),s("mfrac",null,[s("mi",null,"V"),s("mi",null,"R")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"S"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"B"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"\\int \\frac{V}{R} \\mathrm{~d} t=N S \\Delta B / R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.22222em"}},"V")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NS"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(".")])],-1),z=s("h3",{id:"_9-9-静電場のエネルギー",tabindex:"-1"},[a("9.9: 静電場のエネルギー "),s("a",{class:"header-anchor",href:"#_9-9-静電場のエネルギー","aria-label":'Permalink to "9.9: 静電場のエネルギー"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("静電場のエネルギー："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("munder",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"<"),s("mi",null,"j")])]),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"q"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"q"),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"r"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"j")])])]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",null,"∫"),s("mi",null,"ϕ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"q"),s("mo",{separator:"true"},","),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"q"),s("mo",null,"="),s("mi",null,"ρ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"U=k \\sum_{i9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 ​
  1. 一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.

9.11: 全ての電荷 ​

',3),B=s("ol",{start:"12"},[s("li",null,[a("全ての電荷が距離 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" にある場合（例えば，不均一に帯 電した球やリングの中心） "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ϕ"),s("mi",null,"ϕ"),s("mo",null,"="),s("mi",null,"k"),s("mi",null,"Q"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"\\phi \\phi=k Q / r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ϕϕ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord mathnormal"},"Q"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])])])],-1),q=l('

9.12: 外部電荷 ​

  1. 外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.

9.14: 導体 ​

',3),L=s("ol",{start:"14"},[s("li",null,[a("導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"Q")]),s("annotation",{encoding:"application/x-tex"},"Q")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"Q")])])]),a(" という 電荷を持った導電性の球があるように見える).")])],-1),I=s("h3",{id:"_9-15-静電容量",tabindex:"-1"},[a("9.15: 静電容量 "),s("a",{class:"header-anchor",href:"#_9-15-静電容量","aria-label":'Permalink to "9.15: 静電容量"'},"​")],-1),P=s("ol",{start:"15"},[s("li",null,[a("静電容量: "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"C"),s("mo",null,"="),s("mi",null,"ε"),s("mi",null,"S"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"C=\\varepsilon S / d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07153em"}},"C"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"εS"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"d")])])]),a(" (平板), "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"4"),s("mi",null,"π"),s("mi",null,"ε"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"4 \\pi \\varepsilon r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" (球), "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"ε"),s("mi",null,"l"),s("mo",{stretchy:"false"},"("),s("mi",null,"ln"),s("mo",null,"⁡"),s("mi",null,"R"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"1")])])]),s("annotation",{encoding:"application/x-tex"},"2 \\pi \\varepsilon l(\\ln R / r)^{-1}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"εl"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},"ln"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"1")])])])])])])])])])])]),a(" (同軸円筒).")])],-1),S=s("h3",{id:"_9-16-双極子モーメント",tabindex:"-1"},[a("9.16: 双極子モーメント "),s("a",{class:"header-anchor",href:"#_9-16-双極子モーメント","aria-label":'Permalink to "9.16: 双極子モーメント"'},"​")],-1),E=s("ol",{start:"16"},[s("li",null,[s("p",null,"双極子モーメント："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"e")]),s("mo",null,"="),s("mo",null,"∑"),s("msub",null,[s("mi",null,"q"),s("mi",null,"i")]),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"r"),s("mi",null,"i")]),s("mo",null,"="),s("mi",null,"q"),s("mi",{mathvariant:"bold-italic"},"d"),s("mo",{separator:"true"},","),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"μ")]),s("mo",null,"="),s("mi",null,"I"),s("mi",{mathvariant:"bold-italic"},"S")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{p}_e=\\sum q_i \\boldsymbol{r}_i=q \\boldsymbol{d}, \\boldsymbol{p}_\\mu=I \\boldsymbol{S} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6886em","vertical-align":"-0.2441em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2441em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.6em","vertical-align":"-0.55em"}}),s("span",{class:"mop op-symbol large-op",style:{position:"relative",top:"0em"}},"∑"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3117em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0747em","vertical-align":"-0.3802em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"d")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3802em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])])])])])])])])],-1),T=s("h3",{id:"_9-17-双極子場",tabindex:"-1"},[a("9.17: 双極子場 "),s("a",{class:"header-anchor",href:"#_9-17-双極子場","aria-label":'Permalink to "9.17: 双極子場"'},"​")],-1),F=s("ol",{start:"17"},[s("li",null,[a("双極子場 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ϕ"),s("mo",null,"="),s("mi",null,"k"),s("mi",{mathvariant:"bold-italic"},"p"),s("mo",null,"⋅"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"E"),s("mo",{separator:"true"},","),s("mi",null,"B"),s("mo",null,"∝"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"3")])])]),s("annotation",{encoding:"application/x-tex"},"\\phi=k \\boldsymbol{p} \\cdot \\boldsymbol{e}_r / r^2, E, B \\propto r^{-3}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ϕ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"3")])])])])])])])])])])])])],-1),V=s("h3",{id:"_9-18-双極子に働く力",tabindex:"-1"},[a("9.18: 双極子に働く力 "),s("a",{class:"header-anchor",href:"#_9-18-双極子に働く力","aria-label":'Permalink to "9.18: 双極子に働く力"'},"​")],-1),N=s("ol",{start:"19"},[s("li",null,[a("双極子に働く力 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"e")]),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",{fence:"true"},")")]),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")]),s("mo",{separator:"true"},","),s("mi",null,"F"),s("mo",null,"="),s("msup",null,[s("mrow",null,[s("mo",{fence:"true"},"("),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"p"),s("mi",null,"μ")]),s("mo",null,"⋅"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",{fence:"true"},")")]),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"F=\\left(\\boldsymbol{p}_e \\cdot \\boldsymbol{E}\\right)^{\\prime}, F=\\left(\\boldsymbol{p}_\\mu \\cdot \\boldsymbol{B}\\right)^{\\prime}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1418em","vertical-align":"-0.25em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2441em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},")")]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8918em"}},[s("span",{style:{top:"-3.2029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.372em","vertical-align":"-0.3802em"}}),s("span",{class:"minner"},[s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"p")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.0573em"}},[s("span",{style:{top:"-2.4559em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3802em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9918em"}},[s("span",{style:{top:"-3.3029em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(" [訳 者注 : ここの微分はむしろ "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"grad"),s("mo",null,"⁡")]),s("annotation",{encoding:"application/x-tex"},"\\operatorname{grad}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mop"},[s("span",{class:"mord mathrm"},"grad")])])])]),a(" である]. 2 つの双極 子間の相互作用 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("mi",null,"F"),s("mo",null,"∝"),s("msup",null,[s("mi",null,"r"),s("mrow",null,[s("mo",null,"−"),s("mn",null,"4")])])]),s("annotation",{encoding:"application/x-tex"},": F \\propto r^{-4}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"−"),s("span",{class:"mord mtight"},"4")])])])])])])])])])])]),a(".")])],-1),R=s("h3",{id:"_9-19-磁気双極子としての点電荷",tabindex:"-1"},[a("9.19: 磁気双極子としての点電荷 "),s("a",{class:"header-anchor",href:"#_9-19-磁気双極子としての点電荷","aria-label":'Permalink to "9.19: 磁気双極子としての点電荷"'},"​")],-1),C=s("ol",{start:"19"},[s("li",null,[a("磁気双極子としての点電荷 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msub",null,[s("mi",null,"p"),s("mi",null,"μ")]),s("mo",null,"∝"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"∝"),s("msubsup",null,[s("mi",null,"v"),s("mo",{lspace:"0em",rspace:"0em"},"⊥"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"B")]),s("annotation",{encoding:"application/x-tex"},"p_\\mu \\propto \\Phi \\propto v_{\\perp}^2 / B")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7167em","vertical-align":"-0.2861em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"p"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"μ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2861em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0972em","vertical-align":"-0.2831em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-2.4169em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mrel mtight"},"⊥")])])]),s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2831em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B")])])]),a(" は断熱 不変量 ("),s("a",{href:"./4#_4-22-断熱不変量"},"Section 4: #22"),a(" 参照).")])],-1),j=l('

9.20: 鏡像法 ​

  1. 鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.

9.21: 一様（電）場中の球 (円柱) の分極 ​

',3),A=s("ol",{start:"21"},[s("li",null,[a("一様（電）場中の球 (円柱) の分極 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"+"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"(+\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"+"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" と "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"−"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"-\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" に一 様に帯電した球 (円柱) の重ね合わせで， "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d"),s("mo",null,"∝"),s("mi",null,"E")]),s("annotation",{encoding:"application/x-tex"},"d \\propto E")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"∝"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E")])])]),a(".")])],-1),G=s("h3",{id:"_9-22-渦電流",tabindex:"-1"},[a("9.22: 渦電流 "),s("a",{class:"header-anchor",href:"#_9-22-渦電流","aria-label":'Permalink to "9.22: 渦電流"'},"​")],-1),Q=s("ol",{start:"22"},[s("li",null,[a("渦電流: 電流損失密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"≈"),s("msup",null,[s("mi",null,"B"),s("mn",null,"2")]),s("msup",null,[s("mi",null,"v"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"ρ"),s("mi",{mathvariant:"normal"},"."),s("mn",null,"1")]),s("annotation",{encoding:"application/x-tex"},"\\approx B^2 v^2 / \\rho .1")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4831em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"v"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"ρ"),s("span",{class:"mord"},".1")])])]),a(" 回の通過で与え られる運動量 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mi",null,"τ"),s("mo",null,"≈"),s("msup",null,[s("mi",null,"B"),s("mn",null,"2")]),s("msup",null,[s("mi",null,"a"),s("mn",null,"3")]),s("mi",null,"d"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"F \\tau \\approx B^2 a^3 d / \\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.1132em"}},"τ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"≈"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"3")])])])])])])]),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" (ここで "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"d")]),s("annotation",{encoding:"application/x-tex"},"d")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathnormal"},"d")])])]),a(" は厚さ, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"a")]),s("annotation",{encoding:"application/x-tex"},"a")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"a")])])]),a(" は大きさ).")])],-1),U=[i,r,p,c,h,o,g,d,u,y,b,v,x,w,k,_,z,f,M,B,q,L,I,P,S,E,T,F,V,N,R,C,j,A,G,Q];function D(O,$,J,K,H,W){return m(),n("div",null,U)}const Z=t(e,[["render",D]]);export{Y as __pageData,Z as default}; diff --git a/assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.lean.js b/assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.lean.js similarity index 99% rename from assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.lean.js rename to assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.lean.js index 396c604b..b606c1c1 100644 --- a/assets/academic_physics_ipho-formulas-jpn_9.md.6637313d.lean.js +++ b/assets/academic_physics_ipho-formulas-jpn_9.md.fc474d28.lean.js @@ -1 +1 @@ -import{_ as t,c as n,o as m,R as l,z as s,a}from"./chunks/framework.a1bac067.js";const Y=JSON.parse('{"title":"Formulas for IPhO 日本語版: Section 9","description":"","frontmatter":{},"headers":[],"relativePath":"academic/physics/ipho-formulas-jpn/9.md","filePath":"academic/physics/ipho-formulas-jpn/9.md","lastUpdated":1684858945000}'),e={name:"academic/physics/ipho-formulas-jpn/9.md"},i=l("",3),r=s("ol",null,[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"F"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")]),s("mo",{separator:"true"},","),s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("msub",null,[s("mi",null,"q"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"q"),s("mn",null,"2")]),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"r")]),s("annotation",{encoding:"application/x-tex"},"F=k q_1 q_2 / r^2, U=k q_1 q_2 / r")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.13889em"}},"F"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"U"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03148em"}},"k"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3011em"}},[s("span",{style:{top:"-2.55em","margin-left":"-0.0359em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])]),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r")])])]),a(" で, Kepler の法則が 使える ("),s("a",{href:"./12"},"Section 12 参照"),a(").")])],-1),p=s("h3",{id:"_9-2-gauss-の法則",tabindex:"-1"},[a("9.2: Gauss の法則 "),s("a",{class:"header-anchor",href:"#_9-2-gauss-の法則","aria-label":'Permalink to "9.2: Gauss の法則"'},"​")],-1),c=s("ol",{start:"2"},[s("li",null,[s("p",null,[a("Gauss の法則 : "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{S}=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1111em","vertical-align":"-0.3061em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(",")]),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",null,"ε"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mi",null,"Q"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"S"),s("mo",null,"="),s("mo",null,"−"),s("mn",null,"4"),s("mi",null,"π"),s("mi",null,"G"),s("mi",null,"M")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\varepsilon \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{S}=Q, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{S}=-4 \\pi G M ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal"},"Q"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05382em"}},"S")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.10903em"}},"GM")])])])])])])],-1),h=s("h3",{id:"_9-3-循環定理",tabindex:"-1"},[a("9.3: 循環定理 "),s("a",{class:"header-anchor",href:"#_9-3-循環定理","aria-label":'Permalink to "9.3: 循環定理"'},"​")],-1),o=s("ol",{start:"3"},[s("li",null,[s("p",null,"循環定理 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0"),s("mo",{stretchy:"false"},"("),s("mo",null,"="),s("mover",{accent:"true"},[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"˙")]),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l")]),s("mi",null,"μ")]),s("mo",null,"="),s("mi",null,"I"),s("mo",{separator:"true"},","),s("mo",null,"∮"),s("mi",{mathvariant:"bold-italic"},"g"),s("mo",null,"⋅"),s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"\\oint \\boldsymbol{E} \\cdot \\mathrm{d} \\boldsymbol{l}=0(=\\dot{\\Phi}), \\oint \\frac{\\boldsymbol{B} \\cdot \\mathrm{d} \\boldsymbol{l}}{\\mu}=I, \\oint \\boldsymbol{g} \\cdot \\mathrm{d} \\boldsymbol{l}=0 ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"0"),s("span",{class:"mopen"},"("),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2519em","vertical-align":"-0.8804em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9202em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},"Φ")]),s("span",{style:{top:"-3.2523em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.1389em"}},[s("span",{class:"mord"},"˙")])])])])])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2222em","vertical-align":"-0.8622em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mop op-symbol large-op",style:{"margin-right":"0.44445em",position:"relative",top:"-0.0011em"}},"∮"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"g")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"⋅"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])])])])])],-1),g=s("h3",{id:"_9-4-電流素片により生じる磁束密度",tabindex:"-1"},[a("9.4: 電流素片により生じる磁束密度 "),s("a",{class:"header-anchor",href:"#_9-4-電流素片により生じる磁束密度","aria-label":'Permalink to "9.4: 電流素片により生じる磁束密度"'},"​")],-1),d=s("ol",{start:"4"},[s("li",null,[a("電流素片により生じる磁束密度 :"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mfrac",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"d"),s("mi",{mathvariant:"bold-italic"},"l"),s("mo",null,"×"),s("msub",null,[s("mi",{mathvariant:"bold-italic"},"e"),s("mi",null,"r")])]),s("msup",null,[s("mi",null,"r"),s("mn",null,"2")])]),s("mi",{mathvariant:"normal"},".")]),s("annotation",{encoding:"application/x-tex"},"\\mathrm{d} \\boldsymbol{B}=\\frac{\\mu I}{4 \\pi} \\frac{\\mathrm{d} \\boldsymbol{l} \\times \\boldsymbol{e}_r}{r^2} . ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.0574em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3603em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"4"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7401em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"d"),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.0088em"}},"l")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"e")])]),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1514em"}},[s("span",{style:{top:"-2.55em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.15em"}},[s("span")])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},".")])])])])]),a(" したがって電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(" が流れる円形回路の中心では "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(".")])],-1),u=s("h3",{id:"_9-5-ローレンツ力",tabindex:"-1"},[a("9.5: ローレンツ力 "),s("a",{class:"header-anchor",href:"#_9-5-ローレンツ力","aria-label":'Permalink to "9.5: ローレンツ力"'},"​")],-1),y=s("ol",{start:"5"},[s("li",null,[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",null,"e"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"+"),s("mi",{mathvariant:"bold-italic"},"v"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",{stretchy:"false"},")"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"F"),s("mo",null,"="),s("mi",{mathvariant:"bold-italic"},"I"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"B"),s("mi",null,"l")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{F}=e(\\boldsymbol{E}+\\boldsymbol{v} \\times \\boldsymbol{B}), \\boldsymbol{F}=\\boldsymbol{I} \\times \\boldsymbol{B} l")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"e"),s("span",{class:"mopen"},"("),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6667em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03704em"}},"v")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mclose"},")"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.15972em"}},"F")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7694em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.07778em"}},"I")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6944em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.01968em"}},"l")])])]),a(".")])],-1),b=s("h3",{id:"_9-6-gauss-の定理と循環定理より",tabindex:"-1"},[a("9.6: Gauss の定理と循環定理より "),s("a",{class:"header-anchor",href:"#_9-6-gauss-の定理と循環定理より","aria-label":'Permalink to "9.6: Gauss の定理と循環定理より"'},"​")],-1),v=s("ol",{start:"6"},[s("li",null,[a("Gauss の定理と循環定理より：帯電した導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")]),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\pi \\varepsilon_0 r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる導線について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"I")]),s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"r")])])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 I}{2 \\pi r}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 帯電した面について "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"σ"),s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"ε"),s("mn",null,"0")])])])]),s("annotation",{encoding:"application/x-tex"},"E=\\frac{\\sigma}{2 \\varepsilon_0}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.1405em","vertical-align":"-0.4451em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6954em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2"),s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"ε"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"σ")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4451em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(", 電流が流れる面につい て "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"μ"),s("mn",null,"0")]),s("mi",null,"i")]),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"B=\\frac{\\mu_0 i}{2}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2528em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9078em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"2")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3173em"}},[s("span",{style:{top:"-2.357em","margin-left":"0em","margin-right":"0.0714em"}},[s("span",{class:"pstrut",style:{height:"2.5em"}}),s("span",{class:"sizing reset-size3 size1 mtight"},[s("span",{class:"mord mtight"},"0")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.143em"}},[s("span")])])])])]),s("span",{class:"mord mathnormal mtight"},"i")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})])])])]),a(". 一様に帯電した球殼（又は無限に長い円 筒）の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"E"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"E=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"E"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(", 軸に沿って表面に電流が流れる 円筒の内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 密度 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"ρ")]),s("annotation",{encoding:"application/x-tex"},"\\rho")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"ρ")])])]),a(" で一様に帯電, 又は一様 な電流 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"i")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{i}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6933em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])])])])]),a(" が流れる, 球 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"3"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d=3) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"3"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 円柱 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"−"),s("mn",null,"2"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"/")]),s("annotation",{encoding:"application/x-tex"},"(d-2) /")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"2"),s("span",{class:"mclose"},")"),s("span",{class:"mord"},"/")])])]),a(" 平面 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mi",null,"d"),s("mo",null,"="),s("mn",null,"1"),s("mo",{stretchy:"false"},")")]),s("annotation",{encoding:"application/x-tex"},"(d=1)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal"},"d"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")")])])]),a(" の内部で,"),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"bold-italic"},"E"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"ρ"),s("mrow",null,[s("mi",null,"ε"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{separator:"true"},","),s("mi",{mathvariant:"bold-italic"},"B"),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mrow",null,[s("mi",null,"μ"),s("mi",null,"d")])]),s("mi",{mathvariant:"bold-italic"},"i"),s("mo",null,"×"),s("mi",{mathvariant:"bold-italic"},"r")]),s("annotation",{encoding:"application/x-tex"},"\\boldsymbol{E}=\\frac{\\rho}{\\varepsilon d} \\boldsymbol{r}, \\boldsymbol{B}=\\frac{1}{\\mu d} \\boldsymbol{i} \\times \\boldsymbol{r} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6861em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.05451em"}},"E")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.7936em","vertical-align":"-0.686em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.1076em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ε"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"ρ")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])]),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.04835em"}},"B")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.2019em","vertical-align":"-0.8804em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3214em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"d")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"1")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8804em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol"},"i")])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4444em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord boldsymbol",style:{"margin-right":"0.03194em"}},"r")])])])])])])])])],-1),x=s("h3",{id:"_9-7-長いソレノイド",tabindex:"-1"},[a("9.7: 長いソレノイド "),s("a",{class:"header-anchor",href:"#_9-7-長いソレノイド","aria-label":'Permalink to "9.7: 長いソレノイド"'},"​")],-1),w=s("ol",{start:"7"},[s("li",null,[a("長いソレノイド: 内部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I")]),s("annotation",{encoding:"application/x-tex"},"B=\\mu n I")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8778em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I")])])]),a(", 外部で "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"B"),s("mo",null,"="),s("mn",null,"0")]),s("annotation",{encoding:"application/x-tex"},"B=0")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"0")])])]),a(". 磁束 "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"B"),s("mi",null,"S"),s("mrow",null,[s("mo",{fence:"true"},"("),s("mi",null,"n"),s("mo",null,"="),s("mfrac",null,[s("mi",null,"N"),s("mi",null,"l")]),s("mo",{fence:"true"},")")])]),s("annotation",{encoding:"application/x-tex"},"\\Phi=N B S\\left(n=\\frac{N}{l}\\right)")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord"},"Φ"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2223em","vertical-align":"-0.35em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NBS"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},"(")]),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.01968em"}},"l")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.10903em"}},"N")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size1"},")")])])])])]),a(". インダクタンス "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"L"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"L=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6833em"}}),s("span",{class:"mord mathnormal"},"L"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",{mathvariant:"normal"},"Φ"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"I"),s("mo",null,"="),s("mi",null,"μ"),s("msup",null,[s("mi",null,"n"),s("mn",null,"2")]),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"\\Phi / I=\\mu n^2 V")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord"},"Φ/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"μ"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"n"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.22222em"}},"V")])])]),a(". 短いソレノイド "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,":"),s("msub",null,[s("mi",null,"B"),s("mi",{mathvariant:"normal"},"∥")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"μ"),s("mi",null,"n"),s("mi",null,"I"),s("mi",{mathvariant:"normal"},"Ω")]),s("mrow",null,[s("mn",null,"4"),s("mi",null,"π")])]),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"normal"},"Ω")]),s("annotation",{encoding:"application/x-tex"},": B_{\\|}=\\frac{\\mu n I \\Omega}{4 \\pi}(\\Omega")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mrel"},":"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0385em","vertical-align":"-0.3552em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3448em"}},[s("span",{style:{top:"-2.5198em","margin-left":"-0.0502em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"∥")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.3552em"}},[s("span")])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2694em","vertical-align":"-0.345em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.9244em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"4"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"π")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.4461em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"μ"),s("span",{class:"mord mathnormal mtight"},"n"),s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I"),s("span",{class:"mord mtight"},"Ω")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"Ω")])])]),a(" は 立体角).")])],-1),k=s("h3",{id:"_9-8-磁場を小型コイルや衝撃検流計で測定する",tabindex:"-1"},[a("9.8: 磁場を小型コイルや衝撃検流計で測定する "),s("a",{class:"header-anchor",href:"#_9-8-磁場を小型コイルや衝撃検流計で測定する","aria-label":'Permalink to "9.8: 磁場を小型コイルや衝撃検流計で測定する"'},"​")],-1),_=s("ol",{start:"9"},[s("li",null,[a("磁場を小型コイルや衝撃検流計で測定する： "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"q"),s("mo",null,"=")]),s("annotation",{encoding:"application/x-tex"},"q=")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"q"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"=")])])]),a(),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mo",null,"∫"),s("mfrac",null,[s("mi",null,"V"),s("mi",null,"R")]),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"t"),s("mo",null,"="),s("mi",null,"N"),s("mi",null,"S"),s("mi",{mathvariant:"normal"},"Δ"),s("mi",null,"B"),s("mi",{mathvariant:"normal"},"/"),s("mi",null,"R")]),s("annotation",{encoding:"application/x-tex"},"\\int \\frac{V}{R} \\mathrm{~d} t=N S \\Delta B / R")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.2173em","vertical-align":"-0.345em"}}),s("span",{class:"mop op-symbol small-op",style:{"margin-right":"0.19445em",position:"relative",top:"-0.0006em"}},"∫"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8723em"}},[s("span",{style:{top:"-2.655em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.00773em"}},"R")])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.394em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.22222em"}},"V")])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.345em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mord"},[s("span",{class:"mspace nobreak"}," "),s("span",{class:"mord mathrm"},"d")]),s("span",{class:"mord mathnormal"},"t"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05764em"}},"NS"),s("span",{class:"mord"},"Δ"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.05017em"}},"B"),s("span",{class:"mord"},"/"),s("span",{class:"mord mathnormal",style:{"margin-right":"0.00773em"}},"R")])])]),a(".")])],-1),z=s("h3",{id:"_9-9-静電場のエネルギー",tabindex:"-1"},[a("9.9: 静電場のエネルギー "),s("a",{class:"header-anchor",href:"#_9-9-静電場のエネルギー","aria-label":'Permalink to "9.9: 静電場のエネルギー"'},"​")],-1),f=s("ol",{start:"9"},[s("li",null,[a("静電場のエネルギー："),s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"U"),s("mo",null,"="),s("mi",null,"k"),s("munder",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"<"),s("mi",null,"j")])]),s("mfrac",null,[s("mrow",null,[s("msub",null,[s("mi",null,"q"),s("mi",null,"i")]),s("msub",null,[s("mi",null,"q"),s("mi",null,"j")])]),s("msub",null,[s("mi",null,"r"),s("mrow",null,[s("mi",null,"i"),s("mi",null,"j")])])]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("mn",null,"2")]),s("mo",null,"∫"),s("mi",null,"ϕ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"q"),s("mo",{separator:"true"},","),s("mrow",null,[s("mtext",null," "),s("mi",{mathvariant:"normal"},"d")]),s("mi",null,"q"),s("mo",null,"="),s("mi",null,"ρ"),s("mo",{stretchy:"false"},"("),s("mi",{mathvariant:"bold-italic"},"r"),s("mo",{stretchy:"false"},")"),s("mi",{mathvariant:"normal"},"d"),s("mi",null,"V")]),s("annotation",{encoding:"application/x-tex"},"U=k \\sum_{i2023-2-27: Vocabulary ​

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

',4),i=[d];function n(l,s,c,u,p,h){return r(),e("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default}; +import{_ as t,c as e,o as r,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27","frontmatter":{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27"},"headers":[],"relativePath":"academic/vocabulary/2023/02/2023-02-27.md","filePath":"academic/vocabulary/2023/02/2023-02-27.md","lastUpdated":1684859551000}'),a={name:"academic/vocabulary/2023/02/2023-02-27.md"},d=o('

2023-2-27: Vocabulary ​

This table of vocabularies are from "Sadlier Vocabulary Workshop-Level G (Unit 4)" with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.

VocabularyDefinition
Atrophy(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away
BastionA fortified place, stronghold
ConcordA state of agreement, harmony, unanimity; a treaty, pact, covenant
Consummate(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection
Disarray(n.) disorder, confusion; (v.) to throw into disorder
ExigencyUrgency, pressure; urgent demand, pressing need; an emergency
FlotsamFloating debris; homeless, impoverished people
FreneticFrenzied, highly agitated
GleanTo gather bit by bit; to gather small quantities of grain left in a field by the reapers.
Grouse(n.) A type of game bird; a complaint; (v.) to complain, grumble
IncarcerateTo imprison, confine, jail
Incumbent(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of
JocularHumorous, jesting, jolly, joking
LudicrousRidiculous, laughable, absurd
MordantBiting or caustic in thought, manner, or style; sharply or bitterly harsh.
Nettle(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely
PecuniaryConsisting of or measured in money; of or related to money
PusillanimousContemptibly cowardly or mean-spirited
RecumbentIn a reclining position, lying down, in the posture of one sleeping or resting.
StratagemA scheme to outwit or deceive an opponent or to gain and end.

Reference

',4),i=[d];function n(l,s,c,u,p,h){return r(),e("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default}; diff --git a/assets/academic_vocabulary_2023_02_2023-02-27.md.07288c48.lean.js b/assets/academic_vocabulary_2023_02_2023-02-27.md.50b28361.lean.js similarity index 89% rename from assets/academic_vocabulary_2023_02_2023-02-27.md.07288c48.lean.js rename to assets/academic_vocabulary_2023_02_2023-02-27.md.50b28361.lean.js index 019875b8..5860e952 100644 --- a/assets/academic_vocabulary_2023_02_2023-02-27.md.07288c48.lean.js +++ b/assets/academic_vocabulary_2023_02_2023-02-27.md.50b28361.lean.js @@ -1 +1 @@ -import{_ as t,c as e,o as r,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27","frontmatter":{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27"},"headers":[],"relativePath":"academic/vocabulary/2023/02/2023-02-27.md","filePath":"academic/vocabulary/2023/02/2023-02-27.md","lastUpdated":1684858945000}'),a={name:"academic/vocabulary/2023/02/2023-02-27.md"},d=o("",4),i=[d];function n(l,s,c,u,p,h){return r(),e("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default}; +import{_ as t,c as e,o as r,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27","frontmatter":{"title":"2023-2-27: Vocabulary","description":"Vocabulary list collected for 2023-2-27"},"headers":[],"relativePath":"academic/vocabulary/2023/02/2023-02-27.md","filePath":"academic/vocabulary/2023/02/2023-02-27.md","lastUpdated":1684859551000}'),a={name:"academic/vocabulary/2023/02/2023-02-27.md"},d=o("",4),i=[d];function n(l,s,c,u,p,h){return r(),e("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default}; diff --git a/assets/academic_vocabulary_index.md.d46af4f9.js b/assets/academic_vocabulary_index.md.b0a1288c.js similarity index 90% rename from assets/academic_vocabulary_index.md.d46af4f9.js rename to assets/academic_vocabulary_index.md.b0a1288c.js index 24dd69ab..a337ccd7 100644 --- a/assets/academic_vocabulary_index.md.d46af4f9.js +++ b/assets/academic_vocabulary_index.md.b0a1288c.js @@ -1 +1 @@ -import{_ as a,c as t,o,z as e,a as c}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Welcome to My Vocabulary List!","description":"","frontmatter":{},"headers":[],"relativePath":"academic/vocabulary/index.md","filePath":"academic/vocabulary/index.md","lastUpdated":1684858945000}'),r={name:"academic/vocabulary/index.md"},s=e("h1",{id:"welcome-to-my-vocabulary-list",tabindex:"-1"},[c("Welcome to My Vocabulary List! "),e("a",{class:"header-anchor",href:"#welcome-to-my-vocabulary-list","aria-label":'Permalink to "Welcome to My Vocabulary List!"'},"​")],-1),l=[s];function i(d,n,m,_,u,y){return o(),t("div",null,l)}const h=a(r,[["render",i]]);export{b as __pageData,h as default}; +import{_ as a,c as t,o,z as e,a as c}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Welcome to My Vocabulary List!","description":"","frontmatter":{},"headers":[],"relativePath":"academic/vocabulary/index.md","filePath":"academic/vocabulary/index.md","lastUpdated":1684859551000}'),r={name:"academic/vocabulary/index.md"},s=e("h1",{id:"welcome-to-my-vocabulary-list",tabindex:"-1"},[c("Welcome to My Vocabulary List! "),e("a",{class:"header-anchor",href:"#welcome-to-my-vocabulary-list","aria-label":'Permalink to "Welcome to My Vocabulary List!"'},"​")],-1),l=[s];function i(d,n,m,_,u,y){return o(),t("div",null,l)}const h=a(r,[["render",i]]);export{b as __pageData,h as default}; diff --git a/assets/academic_vocabulary_index.md.d46af4f9.lean.js b/assets/academic_vocabulary_index.md.b0a1288c.lean.js similarity index 90% rename from assets/academic_vocabulary_index.md.d46af4f9.lean.js rename to assets/academic_vocabulary_index.md.b0a1288c.lean.js index 24dd69ab..a337ccd7 100644 --- a/assets/academic_vocabulary_index.md.d46af4f9.lean.js +++ b/assets/academic_vocabulary_index.md.b0a1288c.lean.js @@ -1 +1 @@ -import{_ as a,c as t,o,z as e,a as c}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Welcome to My Vocabulary List!","description":"","frontmatter":{},"headers":[],"relativePath":"academic/vocabulary/index.md","filePath":"academic/vocabulary/index.md","lastUpdated":1684858945000}'),r={name:"academic/vocabulary/index.md"},s=e("h1",{id:"welcome-to-my-vocabulary-list",tabindex:"-1"},[c("Welcome to My Vocabulary List! "),e("a",{class:"header-anchor",href:"#welcome-to-my-vocabulary-list","aria-label":'Permalink to "Welcome to My Vocabulary List!"'},"​")],-1),l=[s];function i(d,n,m,_,u,y){return o(),t("div",null,l)}const h=a(r,[["render",i]]);export{b as __pageData,h as default}; +import{_ as a,c as t,o,z as e,a as c}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"Welcome to My Vocabulary List!","description":"","frontmatter":{},"headers":[],"relativePath":"academic/vocabulary/index.md","filePath":"academic/vocabulary/index.md","lastUpdated":1684859551000}'),r={name:"academic/vocabulary/index.md"},s=e("h1",{id:"welcome-to-my-vocabulary-list",tabindex:"-1"},[c("Welcome to My Vocabulary List! "),e("a",{class:"header-anchor",href:"#welcome-to-my-vocabulary-list","aria-label":'Permalink to "Welcome to My Vocabulary List!"'},"​")],-1),l=[s];function i(d,n,m,_,u,y){return o(),t("div",null,l)}const h=a(r,[["render",i]]);export{b as __pageData,h as default}; diff --git a/assets/application_markdown-it-katex_how-to-use.md.ae049342.js b/assets/application_markdown-it-katex_how-to-use.md.b720171d.js similarity index 97% rename from assets/application_markdown-it-katex_how-to-use.md.ae049342.js rename to assets/application_markdown-it-katex_how-to-use.md.b720171d.js index fa7e119e..d91e7889 100644 --- a/assets/application_markdown-it-katex_how-to-use.md.ae049342.js +++ b/assets/application_markdown-it-katex_how-to-use.md.b720171d.js @@ -1,10 +1,10 @@ -import{_ as n,c as e,o as t,z as s,a,R as l}from"./chunks/framework.a1bac067.js";const v=JSON.parse(`{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China","frontmatter":{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China"},"headers":[],"relativePath":"application/markdown-it-katex/how-to-use.md","filePath":"application/markdown-it-katex/how-to-use.md","lastUpdated":1684858945000}`),p={name:"application/markdown-it-katex/how-to-use.md"},o=s("h1",{id:"andatoshiki-markdown-it-katex",tabindex:"-1"},[a("@andatoshiki/markdown-it-katex "),s("a",{class:"header-anchor",href:"#andatoshiki-markdown-it-katex","aria-label":'Permalink to "@andatoshiki/markdown-it-katex"'},"​")],-1),r=s("blockquote",null,[s("p",null,[a("Add graceful "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null,"KaTeX")]),s("annotation",{encoding:"application/x-tex"},"\\KaTeX")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8988em","vertical-align":"-0.2155em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"K"),s("span",{class:"mspace",style:{"margin-right":"-0.17em"}}),s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.905em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm mtight sizing reset-size6 size3"},"A")])])])])]),s("span",{class:"mspace",style:{"margin-right":"-0.15em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"T"),s("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4678em"}},[s("span",{style:{top:"-2.7845em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm"},"E")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2155em"}},[s("span")])])]),s("span",{class:"mspace",style:{"margin-right":"-0.125em"}}),s("span",{class:"mord textrm"},"X")])])])])]),a(" rendering to your Markdown like a charm with "),s("code",null,"markdown-it"),a(" plugin.")])],-1),c=l(`

1: Installation ​

Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.

sh
$ npm install markdown-it --save
$ npm install markdown-it --save

First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.

sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
+import{_ as n,c as e,o as t,z as s,a,R as l}from"./chunks/framework.a1bac067.js";const k=JSON.parse(`{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China","frontmatter":{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China"},"headers":[],"relativePath":"application/markdown-it-katex/how-to-use.md","filePath":"application/markdown-it-katex/how-to-use.md","lastUpdated":1684859551000}`),p={name:"application/markdown-it-katex/how-to-use.md"},o=s("h1",{id:"andatoshiki-markdown-it-katex",tabindex:"-1"},[a("@andatoshiki/markdown-it-katex "),s("a",{class:"header-anchor",href:"#andatoshiki-markdown-it-katex","aria-label":'Permalink to "@andatoshiki/markdown-it-katex"'},"​")],-1),r=s("blockquote",null,[s("p",null,[a("Add graceful "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null,"KaTeX")]),s("annotation",{encoding:"application/x-tex"},"\\KaTeX")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8988em","vertical-align":"-0.2155em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"K"),s("span",{class:"mspace",style:{"margin-right":"-0.17em"}}),s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.905em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm mtight sizing reset-size6 size3"},"A")])])])])]),s("span",{class:"mspace",style:{"margin-right":"-0.15em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"T"),s("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4678em"}},[s("span",{style:{top:"-2.7845em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm"},"E")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2155em"}},[s("span")])])]),s("span",{class:"mspace",style:{"margin-right":"-0.125em"}}),s("span",{class:"mord textrm"},"X")])])])])]),a(" rendering to your Markdown like a charm with "),s("code",null,"markdown-it"),a(" plugin.")])],-1),c=l(`
1: Installation ​
Before you start using this plugin, make sure you have already installed the default markdown-it parser; if not, please run the following command or refer to the official markdown-it documentation.
sh
$ npm install markdown-it --save
$ npm install markdown-it --save
First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing </body> for markdown-it-katex's core utils to be loaded for the static page.
sh
$ npm install -D @andatoshiki/markdown-it-katex
$ npm install -D @andatoshiki/markdown-it-katex
sh
$ yarn add --dev @andatoshiki/markdown-it-katex
$ yarn add --dev @andatoshiki/markdown-it-katex
sh
$ pnpm add -D @andatoshiki/markdown-it-katex
$ pnpm add -D @andatoshiki/markdown-it-katex
html
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
 </body>
<!-- your other body contents ... -->
     <script src="https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js"></script>
-</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
+</body>
Including KaTeX CSS is necessary in the way you are convenient with, either link the stylesheet from a third party CDN into the local HTML <head> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,
Or, you could clone or download the entire repository source of KaTeX and self load the fonts/styles/scripts locally, but if you prefer loading from third-party CDN with faster load speed when deployed to save your server resources, the following CDN links might be your choice, you can always switch to other platforms based on your need.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css" />
html
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css" />
html
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
<link rel="stylesheet" href="https://unpkg.com/katex@0.16.0/dist/katex.min.css" />
scss
// ... your other styles
 @import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
// ... your other styles
-@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
+@import 'https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css';
If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (github-markdown-css) for styling your HTML output with a similar style replica of GitHub's markdown styling to your familiarity.
html
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css" />
This forked project maintained by Anda Toshiki comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version 16.0, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, 16.0 may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.
Warning
Since this project is a fork of the original markdown-it-katex project that hasn't been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around ver. 0.9.0 which clearly is outdated and results in broken styles with overflowing and other potential bug presents.
2: Usage ​
RTo render equations, you need to include the markdown-it-katex plugin in the markdown-it components in your JavaScript or TypeScript file as follows,
js
var md = require('markdown-it')(),
     mk = require('andatoshiki/markdown-it-katex')
 
 md.use(mk)
@@ -47,4 +47,4 @@ M834 80h400000v40h-400000z`})])])])]),s("span",{class:"vlist-s"},"​")]),s("spa
 $$
$$
 \\frac {\\partial^r} {\\partial \\omega^r} \\left(\\frac {y^{\\omega}} {\\omega}\\right)
 = \\left(\\frac {y^{\\omega}} {\\omega}\\right) \\left\\{(\\log y)^r + \\sum_{i=1}^r \\frac {(-1)^ Ir \\cdots (r-i+1) (\\log y)^{ri}} {\\omega^i} \\right\\}
-$$
the above block equation renders the LaTeX equation as a block with output as follows,
`,4),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"∂"),s("mi",null,"r")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"∂"),s("msup",null,[s("mi",null,"ω"),s("mi",null,"r")])])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"{"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"r")]),s("mo",null,"+"),s("munderover",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"r")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"−"),s("mn",null,"1"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"I")]),s("mi",null,"r"),s("mo",null,"⋯"),s("mo",{stretchy:"false"},"("),s("mi",null,"r"),s("mo",null,"−"),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"1"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mi",null,"r"),s("mi",null,"i")])])]),s("msup",null,[s("mi",null,"ω"),s("mi",null,"i")])]),s("mo",{fence:"true"},"}")])]),s("annotation",{encoding:"application/x-tex"},"\\frac {\\partial^r} {\\partial \\omega^r} \\left(\\frac {y^{\\omega}} {\\omega}\\right) = \\left(\\frac {y^{\\omega}} {\\omega}\\right) \\left\\{(\\log y)^r + \\sum_{i=1}^r \\frac {(-1)^ Ir \\cdots (r-i+1) (\\log y)^{ri}} {\\omega^i} \\right\\} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5904em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0277em","vertical-align":"-1.2777em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"{")]),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.6514em"}},[s("span",{style:{top:"-1.8723em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mrel mtight"},"="),s("span",{class:"mord mtight"},"1")])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])]),s("span",{style:{top:"-4.3em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2777em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5183em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7507em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8413em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"}")])])])])])])],-1),d=l('
5: Syntax ​
Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,
Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.
  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.
6: Supported math syntax ​
KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.
Note
Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.
',7),h=[o,r,c,i,m,y,d];function F(u,D,g,B,C,b){return t(),e("div",null,h)}const k=n(p,[["render",F]]);export{v as __pageData,k as default};
+$$

the above block equation renders the LaTeX equation as a block with output as follows,

`,4),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"∂"),s("mi",null,"r")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"∂"),s("msup",null,[s("mi",null,"ω"),s("mi",null,"r")])])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"{"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"r")]),s("mo",null,"+"),s("munderover",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"r")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"−"),s("mn",null,"1"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"I")]),s("mi",null,"r"),s("mo",null,"⋯"),s("mo",{stretchy:"false"},"("),s("mi",null,"r"),s("mo",null,"−"),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"1"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mi",null,"r"),s("mi",null,"i")])])]),s("msup",null,[s("mi",null,"ω"),s("mi",null,"i")])]),s("mo",{fence:"true"},"}")])]),s("annotation",{encoding:"application/x-tex"},"\\frac {\\partial^r} {\\partial \\omega^r} \\left(\\frac {y^{\\omega}} {\\omega}\\right) = \\left(\\frac {y^{\\omega}} {\\omega}\\right) \\left\\{(\\log y)^r + \\sum_{i=1}^r \\frac {(-1)^ Ir \\cdots (r-i+1) (\\log y)^{ri}} {\\omega^i} \\right\\} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5904em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0277em","vertical-align":"-1.2777em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"{")]),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.6514em"}},[s("span",{style:{top:"-1.8723em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mrel mtight"},"="),s("span",{class:"mord mtight"},"1")])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])]),s("span",{style:{top:"-4.3em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2777em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5183em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7507em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8413em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"}")])])])])])])],-1),d=l('

5: Syntax ​

Math parsing in markdown is designed to comply with the "latex-in-markdown" conventions set by Pandoc,

Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.

  • Pandoc. “Pandoc - Pandoc User’s Guide.” Pandoc.org, pandoc.org/MANUAL.html#math. Accessed 2 Mar. 2023.

6: Supported math syntax ​

KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. It is designed to be easy to use, and to provide high-quality mathematical typesetting for web applications, refer to the following pages of the document for the full list of function support in KaTeX.

Note

Due to the large number of equations rendered on the next following pages, it might takes time to entirely load the webpage, please be patient if the webpage seems to not respond; or it could be a result of slow network connection to correctly render all the equation that causes broken formulas, refresh the page to proceed.

',7),h=[o,r,c,i,m,y,d];function F(u,D,g,B,C,b){return t(),e("div",null,h)}const v=n(p,[["render",F]]);export{k as __pageData,v as default}; diff --git a/assets/application_markdown-it-katex_how-to-use.md.ae049342.lean.js b/assets/application_markdown-it-katex_how-to-use.md.b720171d.lean.js similarity index 98% rename from assets/application_markdown-it-katex_how-to-use.md.ae049342.lean.js rename to assets/application_markdown-it-katex_how-to-use.md.b720171d.lean.js index 458e6dac..56a54391 100644 --- a/assets/application_markdown-it-katex_how-to-use.md.ae049342.lean.js +++ b/assets/application_markdown-it-katex_how-to-use.md.b720171d.lean.js @@ -1,4 +1,4 @@ -import{_ as n,c as e,o as t,z as s,a,R as l}from"./chunks/framework.a1bac067.js";const v=JSON.parse(`{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China","frontmatter":{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China"},"headers":[],"relativePath":"application/markdown-it-katex/how-to-use.md","filePath":"application/markdown-it-katex/how-to-use.md","lastUpdated":1684858945000}`),p={name:"application/markdown-it-katex/how-to-use.md"},o=s("h1",{id:"andatoshiki-markdown-it-katex",tabindex:"-1"},[a("@andatoshiki/markdown-it-katex "),s("a",{class:"header-anchor",href:"#andatoshiki-markdown-it-katex","aria-label":'Permalink to "@andatoshiki/markdown-it-katex"'},"​")],-1),r=s("blockquote",null,[s("p",null,[a("Add graceful "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null,"KaTeX")]),s("annotation",{encoding:"application/x-tex"},"\\KaTeX")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8988em","vertical-align":"-0.2155em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"K"),s("span",{class:"mspace",style:{"margin-right":"-0.17em"}}),s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.905em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm mtight sizing reset-size6 size3"},"A")])])])])]),s("span",{class:"mspace",style:{"margin-right":"-0.15em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"T"),s("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4678em"}},[s("span",{style:{top:"-2.7845em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm"},"E")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2155em"}},[s("span")])])]),s("span",{class:"mspace",style:{"margin-right":"-0.125em"}}),s("span",{class:"mord textrm"},"X")])])])])]),a(" rendering to your Markdown like a charm with "),s("code",null,"markdown-it"),a(" plugin.")])],-1),c=l("",23),i=s("p",null,[a("rendered output equation as follows, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mn",null,"3"),s("mi",null,"x"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"+"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{3x-1}+(1+x)^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.04em","vertical-align":"-0.1744em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8656em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord"},"3"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1")])]),s("span",{style:{top:"-2.8256em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 +import{_ as n,c as e,o as t,z as s,a,R as l}from"./chunks/framework.a1bac067.js";const k=JSON.parse(`{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China","frontmatter":{"title":"@andatoshiki/markdown-it-katex","description":"Documentation for Toshiki's markdown-it-katex fork as an alternative for GitHub readme mirror in case site inaccessibility due to SNI interference by GFW for users in China"},"headers":[],"relativePath":"application/markdown-it-katex/how-to-use.md","filePath":"application/markdown-it-katex/how-to-use.md","lastUpdated":1684859551000}`),p={name:"application/markdown-it-katex/how-to-use.md"},o=s("h1",{id:"andatoshiki-markdown-it-katex",tabindex:"-1"},[a("@andatoshiki/markdown-it-katex "),s("a",{class:"header-anchor",href:"#andatoshiki-markdown-it-katex","aria-label":'Permalink to "@andatoshiki/markdown-it-katex"'},"​")],-1),r=s("blockquote",null,[s("p",null,[a("Add graceful "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mtext",null,"KaTeX")]),s("annotation",{encoding:"application/x-tex"},"\\KaTeX")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8988em","vertical-align":"-0.2155em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"K"),s("span",{class:"mspace",style:{"margin-right":"-0.17em"}}),s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6833em"}},[s("span",{style:{top:"-2.905em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm mtight sizing reset-size6 size3"},"A")])])])])]),s("span",{class:"mspace",style:{"margin-right":"-0.15em"}}),s("span",{class:"mord text"},[s("span",{class:"mord textrm"},"T"),s("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.4678em"}},[s("span",{style:{top:"-2.7845em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord textrm"},"E")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.2155em"}},[s("span")])])]),s("span",{class:"mspace",style:{"margin-right":"-0.125em"}}),s("span",{class:"mord textrm"},"X")])])])])]),a(" rendering to your Markdown like a charm with "),s("code",null,"markdown-it"),a(" plugin.")])],-1),c=l("",23),i=s("p",null,[a("rendered output equation as follows, "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msqrt",null,[s("mrow",null,[s("mn",null,"3"),s("mi",null,"x"),s("mo",null,"−"),s("mn",null,"1")])]),s("mo",null,"+"),s("mo",{stretchy:"false"},"("),s("mn",null,"1"),s("mo",null,"+"),s("mi",null,"x"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mn",null,"2")])]),s("annotation",{encoding:"application/x-tex"},"\\sqrt{3x-1}+(1+x)^2")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.04em","vertical-align":"-0.1744em"}}),s("span",{class:"mord sqrt"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8656em"}},[s("span",{class:"svg-align",style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord",style:{"padding-left":"0.833em"}},[s("span",{class:"mord"},"3"),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1")])]),s("span",{style:{top:"-2.8256em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"hide-tail",style:{"min-width":"0.853em",height:"1.08em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"400em",height:"1.08em",viewBox:"0 0 400000 1080",preserveAspectRatio:"xMinYMin slice"},[s("path",{d:`M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 @@ -9,4 +9,4 @@ c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z -M834 80h400000v40h-400000z`})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1744em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(".")],-1),m=l("",4),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"∂"),s("mi",null,"r")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"∂"),s("msup",null,[s("mi",null,"ω"),s("mi",null,"r")])])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"{"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"r")]),s("mo",null,"+"),s("munderover",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"r")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"−"),s("mn",null,"1"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"I")]),s("mi",null,"r"),s("mo",null,"⋯"),s("mo",{stretchy:"false"},"("),s("mi",null,"r"),s("mo",null,"−"),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"1"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mi",null,"r"),s("mi",null,"i")])])]),s("msup",null,[s("mi",null,"ω"),s("mi",null,"i")])]),s("mo",{fence:"true"},"}")])]),s("annotation",{encoding:"application/x-tex"},"\\frac {\\partial^r} {\\partial \\omega^r} \\left(\\frac {y^{\\omega}} {\\omega}\\right) = \\left(\\frac {y^{\\omega}} {\\omega}\\right) \\left\\{(\\log y)^r + \\sum_{i=1}^r \\frac {(-1)^ Ir \\cdots (r-i+1) (\\log y)^{ri}} {\\omega^i} \\right\\} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5904em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0277em","vertical-align":"-1.2777em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"{")]),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.6514em"}},[s("span",{style:{top:"-1.8723em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mrel mtight"},"="),s("span",{class:"mord mtight"},"1")])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])]),s("span",{style:{top:"-4.3em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2777em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5183em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7507em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8413em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"}")])])])])])])],-1),d=l("",7),h=[o,r,c,i,m,y,d];function F(u,D,g,B,C,b){return t(),e("div",null,h)}const k=n(p,[["render",F]]);export{v as __pageData,k as default}; +M834 80h400000v40h-400000z`})])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1744em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord"},"1"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"1.0641em","vertical-align":"-0.25em"}}),s("span",{class:"mord mathnormal"},"x"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"2")])])])])])])])])])]),a(".")],-1),m=l("",4),y=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mfrac",null,[s("msup",null,[s("mi",{mathvariant:"normal"},"∂"),s("mi",null,"r")]),s("mrow",null,[s("mi",{mathvariant:"normal"},"∂"),s("msup",null,[s("mi",null,"ω"),s("mi",null,"r")])])]),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mo",null,"="),s("mrow",null,[s("mo",{fence:"true"},"("),s("mfrac",null,[s("msup",null,[s("mi",null,"y"),s("mi",null,"ω")]),s("mi",null,"ω")]),s("mo",{fence:"true"},")")]),s("mrow",null,[s("mo",{fence:"true"},"{"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"r")]),s("mo",null,"+"),s("munderover",null,[s("mo",null,"∑"),s("mrow",null,[s("mi",null,"i"),s("mo",null,"="),s("mn",null,"1")]),s("mi",null,"r")]),s("mfrac",null,[s("mrow",null,[s("mo",{stretchy:"false"},"("),s("mo",null,"−"),s("mn",null,"1"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mi",null,"I")]),s("mi",null,"r"),s("mo",null,"⋯"),s("mo",{stretchy:"false"},"("),s("mi",null,"r"),s("mo",null,"−"),s("mi",null,"i"),s("mo",null,"+"),s("mn",null,"1"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"("),s("mi",null,"log"),s("mo",null,"⁡"),s("mi",null,"y"),s("msup",null,[s("mo",{stretchy:"false"},")"),s("mrow",null,[s("mi",null,"r"),s("mi",null,"i")])])]),s("msup",null,[s("mi",null,"ω"),s("mi",null,"i")])]),s("mo",{fence:"true"},"}")])]),s("annotation",{encoding:"application/x-tex"},"\\frac {\\partial^r} {\\partial \\omega^r} \\left(\\frac {y^{\\omega}} {\\omega}\\right) = \\left(\\frac {y^{\\omega}} {\\omega}\\right) \\left\\{(\\log y)^r + \\sum_{i=1}^r \\frac {(-1)^ Ir \\cdots (r-i+1) (\\log y)^{ri}} {\\omega^i} \\right\\} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3714em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.5904em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord",style:{"margin-right":"0.05556em"}},"∂"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0277em","vertical-align":"-1.2777em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},"(")]),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.3414em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω")])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6644em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.03588em"}},"ω")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size3"},")")])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},[s("span",{class:"mopen delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"{")]),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mop op-limits"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.6514em"}},[s("span",{style:{top:"-1.8723em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"i"),s("span",{class:"mrel mtight"},"="),s("span",{class:"mord mtight"},"1")])])]),s("span",{style:{top:"-3.05em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",null,[s("span",{class:"mop op-symbol large-op"},"∑")])]),s("span",{style:{top:"-4.3em","margin-left":"0em"}},[s("span",{class:"pstrut",style:{height:"3.05em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2777em"}},[s("span")])])])]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord"},[s("span",{class:"mopen nulldelimiter"}),s("span",{class:"mfrac"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.5183em"}},[s("span",{style:{top:"-2.314em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},[s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"ω"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7507em"}},[s("span",{style:{top:"-2.989em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])]),s("span",{style:{top:"-3.23em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),s("span",{style:{top:"-3.677em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mopen"},"("),s("span",{class:"mord"},"−"),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8413em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.07847em"}},"I")])])])])])])]),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"minner"},"⋯"),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mopen"},"("),s("span",{class:"mord mathnormal",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"−"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"i"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"+"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mclose"},")"),s("span",{class:"mopen"},"("),s("span",{class:"mop"},[a("lo"),s("span",{style:{"margin-right":"0.01389em"}},"g")]),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),s("span",{class:"mclose"},[s("span",{class:"mclose"},")"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8247em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight",style:{"margin-right":"0.02778em"}},"r"),s("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.686em"}},[s("span")])])])]),s("span",{class:"mclose nulldelimiter"})]),s("span",{class:"mclose delimcenter",style:{top:"0em"}},[s("span",{class:"delimsizing size4"},"}")])])])])])])],-1),d=l("",7),h=[o,r,c,i,m,y,d];function F(u,D,g,B,C,b){return t(),e("div",null,h)}const v=n(p,[["render",F]]);export{k as __pageData,v as default}; diff --git a/assets/application_markdown-it-katex_support-function.md.aeaaab84.js b/assets/application_markdown-it-katex_support-function.md.1d1d4129.js similarity index 99% rename from assets/application_markdown-it-katex_support-function.md.aeaaab84.js rename to assets/application_markdown-it-katex_support-function.md.1d1d4129.js index 4e918fd1..7740ec9a 100644 --- a/assets/application_markdown-it-katex_support-function.md.aeaaab84.js +++ b/assets/application_markdown-it-katex_support-function.md.1d1d4129.js @@ -1,4 +1,4 @@ -import{_ as e,c as l,z as s,a,R as t,o as n}from"./chunks/framework.a1bac067.js";const yt=JSON.parse('{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes.","frontmatter":{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes."},"headers":[],"relativePath":"application/markdown-it-katex/support-function.md","filePath":"application/markdown-it-katex/support-function.md","lastUpdated":1684858945000}'),m={name:"application/markdown-it-katex/support-function.md"},i=t('

KaTeX: Supported function ​

note

This is an direct shameful copy of the documentation pulled from KaTeX documentation, I do not own any of the content below on behalf of this part of the documentation, the table might be outdated as versions migrates, please also refer to KaTeX's official documentation.

This following is a list of TeX functions supported by KaTeX. It is sorted into logical groups.

There is a similar Support Table on the next page, sorted alphabetically in am more intuitive way with the lists both supported and un-supported functions, viewing both tables are suggested comprehensibly and interchangeably is highly suggested.

1: Accents ​

',5),c=s("thead",null,[s("tr",null,[s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}})])],-1),p=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"a'")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(),s("code",null,"a'")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"a"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"a")]),s("span",{style:{top:"-3.35em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.25em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(),s("code",null,"\\tilde{a}")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"g"),s("mo",null,"˚")])]),s("annotation",{encoding:"application/x-tex"},"\\mathring{g}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6944em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.3472em"}},[s("span",{class:"mord"},"˚")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(),s("code",null,"\\mathring{g}")])],-1),r=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])])]),s("annotation",{encoding:"application/x-tex"},"a''")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])])])])]),a(),s("code",null,"a''")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mrow",null,[s("mi",null,"a"),s("mi",null,"c")]),s("mo",{stretchy:"true"},"~")])]),s("annotation",{encoding:"application/x-tex"},"\\widetilde{ac}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7166em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7166em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])]),s("span",{class:"svg-align",style:{top:"-3.4306em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{style:{height:"0.286em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.286em",viewBox:"0 0 1033 286",preserveAspectRatio:"none"},[s("path",{d:`M344 55.266c-142 0-300.638 81.316-311.5 86.418 +import{_ as e,c as l,z as s,a,R as t,o as n}from"./chunks/framework.a1bac067.js";const yt=JSON.parse('{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes.","frontmatter":{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes."},"headers":[],"relativePath":"application/markdown-it-katex/support-function.md","filePath":"application/markdown-it-katex/support-function.md","lastUpdated":1684859551000}'),m={name:"application/markdown-it-katex/support-function.md"},i=t('

KaTeX: Supported function ​

note

This is an direct shameful copy of the documentation pulled from KaTeX documentation, I do not own any of the content below on behalf of this part of the documentation, the table might be outdated as versions migrates, please also refer to KaTeX's official documentation.

This following is a list of TeX functions supported by KaTeX. It is sorted into logical groups.

There is a similar Support Table on the next page, sorted alphabetically in am more intuitive way with the lists both supported and un-supported functions, viewing both tables are suggested comprehensibly and interchangeably is highly suggested.

1: Accents ​

',5),c=s("thead",null,[s("tr",null,[s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}})])],-1),p=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"a'")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(),s("code",null,"a'")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"a"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"a")]),s("span",{style:{top:"-3.35em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.25em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(),s("code",null,"\\tilde{a}")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"g"),s("mo",null,"˚")])]),s("annotation",{encoding:"application/x-tex"},"\\mathring{g}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6944em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.3472em"}},[s("span",{class:"mord"},"˚")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(),s("code",null,"\\mathring{g}")])],-1),r=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])])]),s("annotation",{encoding:"application/x-tex"},"a''")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])])])])]),a(),s("code",null,"a''")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mrow",null,[s("mi",null,"a"),s("mi",null,"c")]),s("mo",{stretchy:"true"},"~")])]),s("annotation",{encoding:"application/x-tex"},"\\widetilde{ac}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7166em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7166em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])]),s("span",{class:"svg-align",style:{top:"-3.4306em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{style:{height:"0.286em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.286em",viewBox:"0 0 1033 286",preserveAspectRatio:"none"},[s("path",{d:`M344 55.266c-142 0-300.638 81.316-311.5 86.418 -8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114 c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751 diff --git a/assets/application_markdown-it-katex_support-function.md.aeaaab84.lean.js b/assets/application_markdown-it-katex_support-function.md.1d1d4129.lean.js similarity index 99% rename from assets/application_markdown-it-katex_support-function.md.aeaaab84.lean.js rename to assets/application_markdown-it-katex_support-function.md.1d1d4129.lean.js index e1015225..ca3901bc 100644 --- a/assets/application_markdown-it-katex_support-function.md.aeaaab84.lean.js +++ b/assets/application_markdown-it-katex_support-function.md.1d1d4129.lean.js @@ -1,4 +1,4 @@ -import{_ as e,c as l,z as s,a,R as t,o as n}from"./chunks/framework.a1bac067.js";const yt=JSON.parse('{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes.","frontmatter":{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes."},"headers":[],"relativePath":"application/markdown-it-katex/support-function.md","filePath":"application/markdown-it-katex/support-function.md","lastUpdated":1684858945000}'),m={name:"application/markdown-it-katex/support-function.md"},i=t("",5),c=s("thead",null,[s("tr",null,[s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}})])],-1),p=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"a'")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(),s("code",null,"a'")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"a"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"a")]),s("span",{style:{top:"-3.35em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.25em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(),s("code",null,"\\tilde{a}")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"g"),s("mo",null,"˚")])]),s("annotation",{encoding:"application/x-tex"},"\\mathring{g}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6944em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.3472em"}},[s("span",{class:"mord"},"˚")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(),s("code",null,"\\mathring{g}")])],-1),r=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])])]),s("annotation",{encoding:"application/x-tex"},"a''")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])])])])]),a(),s("code",null,"a''")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mrow",null,[s("mi",null,"a"),s("mi",null,"c")]),s("mo",{stretchy:"true"},"~")])]),s("annotation",{encoding:"application/x-tex"},"\\widetilde{ac}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7166em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7166em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])]),s("span",{class:"svg-align",style:{top:"-3.4306em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{style:{height:"0.286em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.286em",viewBox:"0 0 1033 286",preserveAspectRatio:"none"},[s("path",{d:`M344 55.266c-142 0-300.638 81.316-311.5 86.418 +import{_ as e,c as l,z as s,a,R as t,o as n}from"./chunks/framework.a1bac067.js";const yt=JSON.parse('{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes.","frontmatter":{"title":"KaTeX: Supported functions","description":"Supported functions within KaTeX syntaxes."},"headers":[],"relativePath":"application/markdown-it-katex/support-function.md","filePath":"application/markdown-it-katex/support-function.md","lastUpdated":1684859551000}'),m={name:"application/markdown-it-katex/support-function.md"},i=t("",5),c=s("thead",null,[s("tr",null,[s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}}),s("th",{style:{"text-align":"left"}})])],-1),p=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),s("annotation",{encoding:"application/x-tex"},"a'")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′")])])])])])])])])])])]),a(),s("code",null,"a'")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"a"),s("mo",null,"~")])]),s("annotation",{encoding:"application/x-tex"},"\\tilde{a}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6679em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6679em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal"},"a")]),s("span",{style:{top:"-3.35em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.25em"}},[s("span",{class:"mord"},"~")])])])])])])])])]),a(),s("code",null,"\\tilde{a}")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mi",null,"g"),s("mo",null,"˚")])]),s("annotation",{encoding:"application/x-tex"},"\\mathring{g}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.6944em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"g")]),s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"accent-body",style:{left:"-0.3472em"}},[s("span",{class:"mord"},"˚")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.1944em"}},[s("span")])])])])])])]),a(),s("code",null,"\\mathring{g}")])],-1),r=s("tr",null,[s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mi",null,"a"),s("mrow",null,[s("mo",{mathvariant:"normal"},"′"),s("mo",{mathvariant:"normal"},"′")])])]),s("annotation",{encoding:"application/x-tex"},"a''")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7519em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7519em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"′′")])])])])])])])])])])]),a(),s("code",null,"a''")]),s("td",{style:{"text-align":"left"}},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mover",{accent:"true"},[s("mrow",null,[s("mi",null,"a"),s("mi",null,"c")]),s("mo",{stretchy:"true"},"~")])]),s("annotation",{encoding:"application/x-tex"},"\\widetilde{ac}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7166em"}}),s("span",{class:"mord accent"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7166em"}},[s("span",{style:{top:"-3em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"a"),s("span",{class:"mord mathnormal"},"c")])]),s("span",{class:"svg-align",style:{top:"-3.4306em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{style:{height:"0.286em"}},[s("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"0.286em",viewBox:"0 0 1033 286",preserveAspectRatio:"none"},[s("path",{d:`M344 55.266c-142 0-300.638 81.316-311.5 86.418 -8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114 c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751 diff --git a/assets/application_markdown-it-katex_support-table.md.ff72261b.js b/assets/application_markdown-it-katex_support-table.md.646afbf1.js similarity index 99% rename from assets/application_markdown-it-katex_support-table.md.ff72261b.js rename to assets/application_markdown-it-katex_support-table.md.646afbf1.js index c142ed05..b3c4976b 100644 --- a/assets/application_markdown-it-katex_support-table.md.ff72261b.js +++ b/assets/application_markdown-it-katex_support-table.md.646afbf1.js @@ -1,4 +1,4 @@ -import{_ as e,c as a,z as t,R as n,a as s,o as l}from"./chunks/framework.a1bac067.js";const Za=JSON.parse('{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically.","frontmatter":{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically."},"headers":[],"relativePath":"application/markdown-it-katex/support-table.md","filePath":"application/markdown-it-katex/support-table.md","lastUpdated":1684858945000}'),i={name:"application/markdown-it-katex/support-table.md"},m=n('

KaTeX: Support table ​

note

This is an direct shameful copy of the documentation pulled from KaTeX documentation, I do not own any of the content below on behalf of this part of the documentation, the table might be outdated as versions migrates, please also refer to KaTeX's official documentation.

Symbols ​

',3),c=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"n"),t("mo",{stretchy:"false"},"!")]),t("annotation",{encoding:"application/x-tex"},"n!")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"n"),t("span",{class:"mclose"},"!")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"n!")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," ⁣"),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\!b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\!b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"y"),t("mn",null,"2")])]),t("annotation",{encoding:"application/x-tex"},"\\def\\bar#1{#1^2} \\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8141em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},"2")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\bar#1{#1^2} \\bar{y}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"#")]),t("annotation",{encoding:"application/x-tex"},"\\#")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"#")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"%"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}},[t("code",null,"%this is a comment")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\%"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"%")]),t("annotation",{encoding:"application/x-tex"},"\\%")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord"},"%")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\cr c & d \\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"&")]),t("annotation",{encoding:"application/x-tex"},"\\&")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mrow"),t("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),t("annotation",{encoding:"application/x-tex"},"'")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7519em"}}),t("span",{class:"mord"},[t("span"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7519em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mtight"},"′")])])])])])])])])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\'{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"("),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"(")]),t("annotation",{encoding:"application/x-tex"},"(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"(")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},")"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},")")]),t("annotation",{encoding:"application/x-tex"},")")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},")")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\(…\\)"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mfrac",null,[t("mi",null,"a"),t("mi",null,"b")])])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\(\\frac a b\\)}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6954em"}},[t("span",{style:{top:"-2.655em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.23em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),t("span",{style:{top:"-3.394em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\(\\frac a b\\)}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\ b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace"}," "),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\ b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},'\\"'),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"¨")])]),t("annotation",{encoding:"application/x-tex"},'\\text{\\"{a}}')])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"¨")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,'\\text{\\"{a}}')])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\$"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"$")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\textdollar}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"$")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\,"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\,\\,{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\,\\,{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\."),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˙")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\.{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1389em"}},[t("span",{class:"mord"},"˙")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\.{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\:"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\:\\:{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\:\\:{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\;"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null,"  "),t("mtext",null,"  "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\;\\;{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[s("a"),t("code",null,"\\;\\;{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msub",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x_i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.3117em"}},[t("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.15em"}},[t("span")])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x_i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"_")]),t("annotation",{encoding:"application/x-tex"},"\\_")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em","vertical-align":"-0.31em"}}),t("span",{class:"mord",style:{"margin-right":"0.02778em"}},"_")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},[s("\\"),t("code",null,"`")]),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˋ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\`{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˋ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"<"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"<")]),t("annotation",{encoding:"application/x-tex"},"<")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},"<")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\="),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\={a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5678em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˉ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\={a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},">"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,">")]),t("annotation",{encoding:"application/x-tex"},">")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},">")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\>"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\>\\>{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\>\\>{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"["),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"[")]),t("annotation",{encoding:"application/x-tex"},"[")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"[")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"]"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"]")]),t("annotation",{encoding:"application/x-tex"},"]")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"]")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"{")]),t("annotation",{encoding:"application/x-tex"},"\\{")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"{")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"}")]),t("annotation",{encoding:"application/x-tex"},"\\}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"}")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∣")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∥")]),t("annotation",{encoding:"application/x-tex"},"\\Vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∥")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"no no no breaks")]),t("annotation",{encoding:"application/x-tex"},"\\text{no~no~no~breaks}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"breaks")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{no~no~no~breaks}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˜")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\~{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"˜")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\~{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x^i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8247em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8247em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x^i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˆ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\^{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˆ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\^{a}}")])])])],-1),p=t("h2",{id:"a",tabindex:"-1"},[s("A "),t("a",{class:"header-anchor",href:"#a","aria-label":'Permalink to "A"'},"​")],-1),r=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AA"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"A"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AA}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.9468em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.9468em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"A")]),t("span",{style:{top:"-3.2523em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AA}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aa"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\aa}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"a")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\aa}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\above"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0.2em"},[t("mi",null,"a"),t("mrow",null,[t("mi",null,"b"),t("mo",null,"+"),t("mn",null,"1")])])]),t("annotation",{encoding:"application/x-tex"},"{a \\above{2pt} b+1}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.4458em","vertical-align":"-0.5944em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8514em"}},[t("span",{style:{top:"-2.4639em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b"),t("span",{class:"mbin mtight"},"+"),t("span",{class:"mord mtight"},"1")])])]),t("span",{style:{top:"-3.15em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.2em"}})]),t("span",{style:{top:"-3.55em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5944em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\above{2pt} b+1}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\abovewithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\acute"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"e"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\acute e")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal"},"e")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\acute e")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AE"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"Æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AE}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"Æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AE}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ae"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\ae}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\ae}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alef"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alef")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alefsym"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alefsym")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aleph"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\aleph")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},align:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},aligned:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"b"),t("mo",null,"+"),t("mi",null,"c")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mi",null,"d"),t("mo",null,"+"),t("mi",null,"e")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"f")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{aligned}a&=b+c\\\\d+e&=f\\end{aligned}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"e")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal"},"b"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"c")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{aligned}"),t("br"),s("   "),t("code",null,"a&=b+c \\\\"),t("br"),s("   "),t("code",null,"d+e&=f"),t("br"),t("code",null,"\\end{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignat:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignedat:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"10")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"2")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"13")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"4")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{alignedat}{2}10&x+&3&y=2\\\\3&x+&13&y=4\\end{alignedat}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"10")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"13")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"2")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"4")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{alignedat}{2}"),t("br"),s("   "),t("code",null,"10&x+ &3&y = 2 \\\\"),t("br"),s("   "),t("code",null," 3&x+&13&y = 4"),t("br"),t("code",null,"\\end{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\allowbreak"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"A")]),t("annotation",{encoding:"application/x-tex"},"\\Alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"A")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"α")]),t("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\amalg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⨿")]),t("annotation",{encoding:"application/x-tex"},"\\amalg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord"},"⨿")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\And"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"&")]),t("annotation",{encoding:"application/x-tex"},"\\And")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\and"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ang"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angl"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angle"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∠")]),t("annotation",{encoding:"application/x-tex"},"\\angle")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mord"},"∠")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approx"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≈")]),t("annotation",{encoding:"application/x-tex"},"\\approx")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4831em"}}),t("span",{class:"mrel"},"≈")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approxeq"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≊")]),t("annotation",{encoding:"application/x-tex"},"\\approxeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6633em","vertical-align":"-0.0817em"}}),t("span",{class:"mrel amsrm"},"≊")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arccos"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arccos"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arccos")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mop"},"arccos")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arcct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcsin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcsin"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcsin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mop"},"arcsin")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctan"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctan"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctan")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6151em"}}),t("span",{class:"mop"},"arctan")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("ar"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmax"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg max"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmax")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"max")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg min"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"min")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},array:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\array"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arraystretch"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.66em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\def\\arraystretch{1.5}\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3.6em","vertical-align":"-1.55em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\arraystretch{1.5}"),t("br"),t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ast"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∗")]),t("annotation",{encoding:"application/x-tex"},"\\ast")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4653em"}}),t("span",{class:"mord"},"∗")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\asymp"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≍")]),t("annotation",{encoding:"application/x-tex"},"\\asymp")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4637em"}}),t("span",{class:"mrel"},"≍")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atop"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0px"},[t("mi",null,"a"),t("mi",null,"b")])]),t("annotation",{encoding:"application/x-tex"},"{a \\atop b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0904em","vertical-align":"-0.345em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7454em"}},[t("span",{style:{top:"-2.355em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.144em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\atop b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atopwithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})])])],-1),h=t("h2",{id:"b",tabindex:"-1"},[s("B "),t("a",{class:"header-anchor",href:"#b","aria-label":'Permalink to "B"'},"​")],-1),o=t("p",null,[s("| Symbol/Function | Rendered | Source or Comment | | :------------------ | :-------------------------------------------------- | :------------------------------------------------------------------------------------------------ | ------ | | \\backepsilon | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∍")]),t("annotation",{encoding:"application/x-tex"},"\\backepsilon")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mrel amsrm"},"∍")])])]),s(" | | | \\backprime | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"‵")]),t("annotation",{encoding:"application/x-tex"},"\\backprime")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5499em"}}),t("span",{class:"mord amsrm"},"‵")])])]),s(" | | | \\backsim | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∽")]),t("annotation",{encoding:"application/x-tex"},"\\backsim")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.3779em"}}),t("span",{class:"mrel amsrm"},"∽")])])]),s(" | | | \\backsimeq | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋍")]),t("annotation",{encoding:"application/x-tex"},"\\backsimeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.464em"}}),t("span",{class:"mrel amsrm"},"⋍")])])]),s(" | | | \\backslash | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"\\")]),t("annotation",{encoding:"application/x-tex"},"\\backslash")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"\\")])])]),s(" | | | \\bar | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"y"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7622em","vertical-align":"-0.1944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˉ")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.1944em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bar{y}"),s(" | | \\barwedge | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⊼")]),t("annotation",{encoding:"application/x-tex"},"\\barwedge")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8867em","vertical-align":"-0.1944em"}}),t("span",{class:"mord amsrm"},"⊼")])])]),s(" | | | \\Bbb | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"A"),t("mi",{mathvariant:"double-struck"},"B"),t("mi",{mathvariant:"double-struck"},"C")]),t("annotation",{encoding:"application/x-tex"},"\\Bbb{ABC}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbb"},"ABC")])])])]),s(" | "),t("code",null,"\\Bbb{ABC}"),t("br"),s("KaTeX supports A-Z & k | | \\Bbbk | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"k")]),t("annotation",{encoding:"application/x-tex"},"\\Bbbk")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord mathbb"},"k")])])]),s(" | | | \\bbox | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\bcancel | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("menclose",{notation:"downdiagonalstrike"},[t("mn",null,"5")])]),t("annotation",{encoding:"application/x-tex"},"\\bcancel{5}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6444em"}}),t("span",{class:"mord"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8444em"}},[t("span",{style:{top:"-3.0444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"5")])]),t("span",{class:"svg-align",style:{top:"-2.8444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{style:{height:"1.0444em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"1.0444em"},[t("line",{x1:"0",y1:"0",x2:"100%",y2:"100%","stroke-width":"0.046em"})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.2em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bcancel{5}"),s(" | | \\because | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∵")]),t("annotation",{encoding:"application/x-tex"},"\\because")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mrel amsrm"},"∵")])])]),s(" | | | \\begin | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])]),s(" | "),t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}"),s(" | | \\begingroup | "),t("span",{class:"katex-error",title:"ParseError: KaTeX parse error: Expected '\\endgroup', got '}' at position 14: \\begingroup a}̲"},"\\begingroup a}"),s(" | "),t("code",null,"\\begingroup a}"),s(" | | \\Beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"B")]),t("annotation",{encoding:"application/x-tex"},"\\Beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"B")])])]),s(" | | | \\beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"β")]),t("annotation",{encoding:"application/x-tex"},"\\beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s(" | | | \\beth | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℶ")]),t("annotation",{encoding:"application/x-tex"},"\\beth")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord amsrm"},"ℶ")])])]),s(" | | | \\between | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≬")]),t("annotation",{encoding:"application/x-tex"},"\\between")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0117em","vertical-align":"-0.2558em"}}),t("span",{class:"mrel amsrm"},"≬")])])]),s(" | | | \\bf | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"bold"},"A"),t("mi",{mathvariant:"bold"},"a"),t("mi",{mathvariant:"bold"},"B"),t("mi",{mathvariant:"bold"},"b"),t("mn",{mathvariant:"bold"},"12")]),t("annotation",{encoding:"application/x-tex"},"\\bf AaBb12")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbf"},"AaBb12")])])])]),s(" | "),t("code",null,"\\bf AaBb12"),s(" | | \\bfseries | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\big(\\big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},")")])])])]),s(" | "),t("code",null,"\\big(\\big)"),s(" | | \\Big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Big(\\Big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},")")])])])]),s(" | "),t("code",null,"\\Big(\\Big)"),s(" | | \\bigcap | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋂")]),t("annotation",{encoding:"application/x-tex"},"\\bigcap")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋂")])])]),s(" | | | \\bigcirc | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"◯")]),t("annotation",{encoding:"application/x-tex"},"\\bigcirc")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"◯")])])]),s(" | | | \\bigcup | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋃")]),t("annotation",{encoding:"application/x-tex"},"\\bigcup")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋃")])])]),s(" | | | \\bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\bigg(\\bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},")")])])])]),s(" | "),t("code",null,"\\bigg(\\bigg)"),s(" | | \\Bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Bigg(\\Bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},")")])])])]),s(" | "),t("code",null,"\\Bigg(\\Bigg)"),s(" | | \\biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size3"},"(")])])])]),s(" | "),t("code",null,"\\biggl("),s(" | | \\Biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"3em",maxsize:"3em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\Biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size4"},"(")])])])]),s(" | "),t("code",null,"\\Biggl("),s(" | | \\biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.45em"}},[t("span",{class:"pstrut",style:{height:"4.4em"}}),t("span",{style:{width:"0.333em",height:"2.400em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"2.400em",viewBox:"0 0 333 2400"},[t("path",{d:`M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15 +import{_ as e,c as a,z as t,R as n,a as s,o as l}from"./chunks/framework.a1bac067.js";const Za=JSON.parse('{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically.","frontmatter":{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically."},"headers":[],"relativePath":"application/markdown-it-katex/support-table.md","filePath":"application/markdown-it-katex/support-table.md","lastUpdated":1684859551000}'),i={name:"application/markdown-it-katex/support-table.md"},m=n('

KaTeX: Support table ​

note

This is an direct shameful copy of the documentation pulled from KaTeX documentation, I do not own any of the content below on behalf of this part of the documentation, the table might be outdated as versions migrates, please also refer to KaTeX's official documentation.

Symbols ​

',3),c=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"n"),t("mo",{stretchy:"false"},"!")]),t("annotation",{encoding:"application/x-tex"},"n!")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"n"),t("span",{class:"mclose"},"!")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"n!")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," ⁣"),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\!b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\!b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"y"),t("mn",null,"2")])]),t("annotation",{encoding:"application/x-tex"},"\\def\\bar#1{#1^2} \\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8141em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},"2")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\bar#1{#1^2} \\bar{y}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"#")]),t("annotation",{encoding:"application/x-tex"},"\\#")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"#")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"%"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}},[t("code",null,"%this is a comment")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\%"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"%")]),t("annotation",{encoding:"application/x-tex"},"\\%")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord"},"%")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\cr c & d \\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"&")]),t("annotation",{encoding:"application/x-tex"},"\\&")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mrow"),t("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),t("annotation",{encoding:"application/x-tex"},"'")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7519em"}}),t("span",{class:"mord"},[t("span"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7519em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mtight"},"′")])])])])])])])])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\'{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"("),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"(")]),t("annotation",{encoding:"application/x-tex"},"(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"(")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},")"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},")")]),t("annotation",{encoding:"application/x-tex"},")")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},")")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\(…\\)"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mfrac",null,[t("mi",null,"a"),t("mi",null,"b")])])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\(\\frac a b\\)}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6954em"}},[t("span",{style:{top:"-2.655em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.23em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),t("span",{style:{top:"-3.394em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\(\\frac a b\\)}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\ b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace"}," "),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\ b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},'\\"'),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"¨")])]),t("annotation",{encoding:"application/x-tex"},'\\text{\\"{a}}')])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"¨")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,'\\text{\\"{a}}')])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\$"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"$")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\textdollar}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"$")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\,"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\,\\,{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\,\\,{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\."),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˙")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\.{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1389em"}},[t("span",{class:"mord"},"˙")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\.{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\:"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\:\\:{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\:\\:{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\;"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null,"  "),t("mtext",null,"  "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\;\\;{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[s("a"),t("code",null,"\\;\\;{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msub",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x_i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.3117em"}},[t("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.15em"}},[t("span")])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x_i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"_")]),t("annotation",{encoding:"application/x-tex"},"\\_")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em","vertical-align":"-0.31em"}}),t("span",{class:"mord",style:{"margin-right":"0.02778em"}},"_")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},[s("\\"),t("code",null,"`")]),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˋ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\`{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˋ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"<"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"<")]),t("annotation",{encoding:"application/x-tex"},"<")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},"<")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\="),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\={a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5678em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˉ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\={a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},">"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,">")]),t("annotation",{encoding:"application/x-tex"},">")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},">")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\>"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\>\\>{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\>\\>{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"["),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"[")]),t("annotation",{encoding:"application/x-tex"},"[")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"[")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"]"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"]")]),t("annotation",{encoding:"application/x-tex"},"]")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"]")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"{")]),t("annotation",{encoding:"application/x-tex"},"\\{")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"{")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"}")]),t("annotation",{encoding:"application/x-tex"},"\\}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"}")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∣")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∥")]),t("annotation",{encoding:"application/x-tex"},"\\Vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∥")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"no no no breaks")]),t("annotation",{encoding:"application/x-tex"},"\\text{no~no~no~breaks}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"breaks")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{no~no~no~breaks}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˜")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\~{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"˜")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\~{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x^i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8247em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8247em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x^i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˆ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\^{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˆ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\^{a}}")])])])],-1),p=t("h2",{id:"a",tabindex:"-1"},[s("A "),t("a",{class:"header-anchor",href:"#a","aria-label":'Permalink to "A"'},"​")],-1),r=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AA"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"A"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AA}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.9468em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.9468em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"A")]),t("span",{style:{top:"-3.2523em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AA}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aa"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\aa}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"a")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\aa}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\above"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0.2em"},[t("mi",null,"a"),t("mrow",null,[t("mi",null,"b"),t("mo",null,"+"),t("mn",null,"1")])])]),t("annotation",{encoding:"application/x-tex"},"{a \\above{2pt} b+1}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.4458em","vertical-align":"-0.5944em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8514em"}},[t("span",{style:{top:"-2.4639em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b"),t("span",{class:"mbin mtight"},"+"),t("span",{class:"mord mtight"},"1")])])]),t("span",{style:{top:"-3.15em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.2em"}})]),t("span",{style:{top:"-3.55em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5944em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\above{2pt} b+1}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\abovewithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\acute"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"e"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\acute e")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal"},"e")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\acute e")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AE"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"Æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AE}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"Æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AE}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ae"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\ae}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\ae}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alef"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alef")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alefsym"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alefsym")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aleph"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\aleph")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},align:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},aligned:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"b"),t("mo",null,"+"),t("mi",null,"c")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mi",null,"d"),t("mo",null,"+"),t("mi",null,"e")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"f")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{aligned}a&=b+c\\\\d+e&=f\\end{aligned}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"e")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal"},"b"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"c")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{aligned}"),t("br"),s("   "),t("code",null,"a&=b+c \\\\"),t("br"),s("   "),t("code",null,"d+e&=f"),t("br"),t("code",null,"\\end{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignat:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignedat:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"10")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"2")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"13")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"4")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{alignedat}{2}10&x+&3&y=2\\\\3&x+&13&y=4\\end{alignedat}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"10")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"13")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"2")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"4")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{alignedat}{2}"),t("br"),s("   "),t("code",null,"10&x+ &3&y = 2 \\\\"),t("br"),s("   "),t("code",null," 3&x+&13&y = 4"),t("br"),t("code",null,"\\end{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\allowbreak"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"A")]),t("annotation",{encoding:"application/x-tex"},"\\Alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"A")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"α")]),t("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\amalg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⨿")]),t("annotation",{encoding:"application/x-tex"},"\\amalg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord"},"⨿")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\And"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"&")]),t("annotation",{encoding:"application/x-tex"},"\\And")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\and"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ang"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angl"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angle"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∠")]),t("annotation",{encoding:"application/x-tex"},"\\angle")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mord"},"∠")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approx"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≈")]),t("annotation",{encoding:"application/x-tex"},"\\approx")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4831em"}}),t("span",{class:"mrel"},"≈")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approxeq"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≊")]),t("annotation",{encoding:"application/x-tex"},"\\approxeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6633em","vertical-align":"-0.0817em"}}),t("span",{class:"mrel amsrm"},"≊")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arccos"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arccos"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arccos")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mop"},"arccos")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arcct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcsin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcsin"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcsin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mop"},"arcsin")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctan"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctan"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctan")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6151em"}}),t("span",{class:"mop"},"arctan")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("ar"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmax"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg max"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmax")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"max")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg min"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"min")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},array:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\array"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arraystretch"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.66em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\def\\arraystretch{1.5}\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3.6em","vertical-align":"-1.55em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\arraystretch{1.5}"),t("br"),t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ast"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∗")]),t("annotation",{encoding:"application/x-tex"},"\\ast")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4653em"}}),t("span",{class:"mord"},"∗")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\asymp"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≍")]),t("annotation",{encoding:"application/x-tex"},"\\asymp")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4637em"}}),t("span",{class:"mrel"},"≍")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atop"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0px"},[t("mi",null,"a"),t("mi",null,"b")])]),t("annotation",{encoding:"application/x-tex"},"{a \\atop b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0904em","vertical-align":"-0.345em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7454em"}},[t("span",{style:{top:"-2.355em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.144em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\atop b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atopwithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})])])],-1),h=t("h2",{id:"b",tabindex:"-1"},[s("B "),t("a",{class:"header-anchor",href:"#b","aria-label":'Permalink to "B"'},"​")],-1),o=t("p",null,[s("| Symbol/Function | Rendered | Source or Comment | | :------------------ | :-------------------------------------------------- | :------------------------------------------------------------------------------------------------ | ------ | | \\backepsilon | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∍")]),t("annotation",{encoding:"application/x-tex"},"\\backepsilon")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mrel amsrm"},"∍")])])]),s(" | | | \\backprime | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"‵")]),t("annotation",{encoding:"application/x-tex"},"\\backprime")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5499em"}}),t("span",{class:"mord amsrm"},"‵")])])]),s(" | | | \\backsim | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∽")]),t("annotation",{encoding:"application/x-tex"},"\\backsim")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.3779em"}}),t("span",{class:"mrel amsrm"},"∽")])])]),s(" | | | \\backsimeq | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋍")]),t("annotation",{encoding:"application/x-tex"},"\\backsimeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.464em"}}),t("span",{class:"mrel amsrm"},"⋍")])])]),s(" | | | \\backslash | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"\\")]),t("annotation",{encoding:"application/x-tex"},"\\backslash")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"\\")])])]),s(" | | | \\bar | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"y"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7622em","vertical-align":"-0.1944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˉ")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.1944em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bar{y}"),s(" | | \\barwedge | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⊼")]),t("annotation",{encoding:"application/x-tex"},"\\barwedge")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8867em","vertical-align":"-0.1944em"}}),t("span",{class:"mord amsrm"},"⊼")])])]),s(" | | | \\Bbb | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"A"),t("mi",{mathvariant:"double-struck"},"B"),t("mi",{mathvariant:"double-struck"},"C")]),t("annotation",{encoding:"application/x-tex"},"\\Bbb{ABC}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbb"},"ABC")])])])]),s(" | "),t("code",null,"\\Bbb{ABC}"),t("br"),s("KaTeX supports A-Z & k | | \\Bbbk | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"k")]),t("annotation",{encoding:"application/x-tex"},"\\Bbbk")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord mathbb"},"k")])])]),s(" | | | \\bbox | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\bcancel | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("menclose",{notation:"downdiagonalstrike"},[t("mn",null,"5")])]),t("annotation",{encoding:"application/x-tex"},"\\bcancel{5}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6444em"}}),t("span",{class:"mord"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8444em"}},[t("span",{style:{top:"-3.0444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"5")])]),t("span",{class:"svg-align",style:{top:"-2.8444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{style:{height:"1.0444em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"1.0444em"},[t("line",{x1:"0",y1:"0",x2:"100%",y2:"100%","stroke-width":"0.046em"})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.2em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bcancel{5}"),s(" | | \\because | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∵")]),t("annotation",{encoding:"application/x-tex"},"\\because")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mrel amsrm"},"∵")])])]),s(" | | | \\begin | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])]),s(" | "),t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}"),s(" | | \\begingroup | "),t("span",{class:"katex-error",title:"ParseError: KaTeX parse error: Expected '\\endgroup', got '}' at position 14: \\begingroup a}̲"},"\\begingroup a}"),s(" | "),t("code",null,"\\begingroup a}"),s(" | | \\Beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"B")]),t("annotation",{encoding:"application/x-tex"},"\\Beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"B")])])]),s(" | | | \\beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"β")]),t("annotation",{encoding:"application/x-tex"},"\\beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s(" | | | \\beth | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℶ")]),t("annotation",{encoding:"application/x-tex"},"\\beth")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord amsrm"},"ℶ")])])]),s(" | | | \\between | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≬")]),t("annotation",{encoding:"application/x-tex"},"\\between")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0117em","vertical-align":"-0.2558em"}}),t("span",{class:"mrel amsrm"},"≬")])])]),s(" | | | \\bf | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"bold"},"A"),t("mi",{mathvariant:"bold"},"a"),t("mi",{mathvariant:"bold"},"B"),t("mi",{mathvariant:"bold"},"b"),t("mn",{mathvariant:"bold"},"12")]),t("annotation",{encoding:"application/x-tex"},"\\bf AaBb12")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbf"},"AaBb12")])])])]),s(" | "),t("code",null,"\\bf AaBb12"),s(" | | \\bfseries | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\big(\\big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},")")])])])]),s(" | "),t("code",null,"\\big(\\big)"),s(" | | \\Big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Big(\\Big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},")")])])])]),s(" | "),t("code",null,"\\Big(\\Big)"),s(" | | \\bigcap | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋂")]),t("annotation",{encoding:"application/x-tex"},"\\bigcap")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋂")])])]),s(" | | | \\bigcirc | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"◯")]),t("annotation",{encoding:"application/x-tex"},"\\bigcirc")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"◯")])])]),s(" | | | \\bigcup | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋃")]),t("annotation",{encoding:"application/x-tex"},"\\bigcup")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋃")])])]),s(" | | | \\bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\bigg(\\bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},")")])])])]),s(" | "),t("code",null,"\\bigg(\\bigg)"),s(" | | \\Bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Bigg(\\Bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},")")])])])]),s(" | "),t("code",null,"\\Bigg(\\Bigg)"),s(" | | \\biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size3"},"(")])])])]),s(" | "),t("code",null,"\\biggl("),s(" | | \\Biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"3em",maxsize:"3em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\Biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size4"},"(")])])])]),s(" | "),t("code",null,"\\Biggl("),s(" | | \\biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.45em"}},[t("span",{class:"pstrut",style:{height:"4.4em"}}),t("span",{style:{width:"0.333em",height:"2.400em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"2.400em",viewBox:"0 0 333 2400"},[t("path",{d:`M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15 c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15 c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z`})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])]),s(" | "),t("code",null,"\\biggm\\vert"),s(" | | \\Biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\Biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.75em"}},[t("span",{class:"pstrut",style:{height:"5em"}}),t("span",{style:{width:"0.333em",height:"3.000em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"3.000em",viewBox:"0 0 333 3000"},[t("path",{d:`M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15 c10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15 diff --git a/assets/application_markdown-it-katex_support-table.md.ff72261b.lean.js b/assets/application_markdown-it-katex_support-table.md.646afbf1.lean.js similarity index 99% rename from assets/application_markdown-it-katex_support-table.md.ff72261b.lean.js rename to assets/application_markdown-it-katex_support-table.md.646afbf1.lean.js index 667f9c7d..cb9c10f2 100644 --- a/assets/application_markdown-it-katex_support-table.md.ff72261b.lean.js +++ b/assets/application_markdown-it-katex_support-table.md.646afbf1.lean.js @@ -1,4 +1,4 @@ -import{_ as e,c as a,z as t,R as n,a as s,o as l}from"./chunks/framework.a1bac067.js";const Za=JSON.parse('{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically.","frontmatter":{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically."},"headers":[],"relativePath":"application/markdown-it-katex/support-table.md","filePath":"application/markdown-it-katex/support-table.md","lastUpdated":1684858945000}'),i={name:"application/markdown-it-katex/support-table.md"},m=n("",3),c=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"n"),t("mo",{stretchy:"false"},"!")]),t("annotation",{encoding:"application/x-tex"},"n!")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"n"),t("span",{class:"mclose"},"!")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"n!")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," ⁣"),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\!b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\!b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"y"),t("mn",null,"2")])]),t("annotation",{encoding:"application/x-tex"},"\\def\\bar#1{#1^2} \\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8141em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},"2")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\bar#1{#1^2} \\bar{y}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"#")]),t("annotation",{encoding:"application/x-tex"},"\\#")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"#")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"%"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}},[t("code",null,"%this is a comment")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\%"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"%")]),t("annotation",{encoding:"application/x-tex"},"\\%")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord"},"%")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\cr c & d \\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"&")]),t("annotation",{encoding:"application/x-tex"},"\\&")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mrow"),t("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),t("annotation",{encoding:"application/x-tex"},"'")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7519em"}}),t("span",{class:"mord"},[t("span"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7519em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mtight"},"′")])])])])])])])])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\'{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"("),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"(")]),t("annotation",{encoding:"application/x-tex"},"(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"(")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},")"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},")")]),t("annotation",{encoding:"application/x-tex"},")")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},")")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\(…\\)"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mfrac",null,[t("mi",null,"a"),t("mi",null,"b")])])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\(\\frac a b\\)}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6954em"}},[t("span",{style:{top:"-2.655em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.23em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),t("span",{style:{top:"-3.394em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\(\\frac a b\\)}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\ b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace"}," "),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\ b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},'\\"'),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"¨")])]),t("annotation",{encoding:"application/x-tex"},'\\text{\\"{a}}')])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"¨")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,'\\text{\\"{a}}')])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\$"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"$")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\textdollar}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"$")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\,"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\,\\,{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\,\\,{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\."),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˙")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\.{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1389em"}},[t("span",{class:"mord"},"˙")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\.{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\:"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\:\\:{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\:\\:{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\;"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null,"  "),t("mtext",null,"  "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\;\\;{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[s("a"),t("code",null,"\\;\\;{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msub",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x_i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.3117em"}},[t("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.15em"}},[t("span")])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x_i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"_")]),t("annotation",{encoding:"application/x-tex"},"\\_")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em","vertical-align":"-0.31em"}}),t("span",{class:"mord",style:{"margin-right":"0.02778em"}},"_")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},[s("\\"),t("code",null,"`")]),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˋ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\`{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˋ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"<"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"<")]),t("annotation",{encoding:"application/x-tex"},"<")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},"<")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\="),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\={a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5678em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˉ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\={a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},">"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,">")]),t("annotation",{encoding:"application/x-tex"},">")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},">")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\>"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\>\\>{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\>\\>{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"["),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"[")]),t("annotation",{encoding:"application/x-tex"},"[")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"[")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"]"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"]")]),t("annotation",{encoding:"application/x-tex"},"]")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"]")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"{")]),t("annotation",{encoding:"application/x-tex"},"\\{")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"{")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"}")]),t("annotation",{encoding:"application/x-tex"},"\\}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"}")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∣")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∥")]),t("annotation",{encoding:"application/x-tex"},"\\Vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∥")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"no no no breaks")]),t("annotation",{encoding:"application/x-tex"},"\\text{no~no~no~breaks}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"breaks")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{no~no~no~breaks}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˜")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\~{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"˜")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\~{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x^i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8247em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8247em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x^i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˆ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\^{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˆ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\^{a}}")])])])],-1),p=t("h2",{id:"a",tabindex:"-1"},[s("A "),t("a",{class:"header-anchor",href:"#a","aria-label":'Permalink to "A"'},"​")],-1),r=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AA"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"A"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AA}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.9468em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.9468em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"A")]),t("span",{style:{top:"-3.2523em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AA}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aa"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\aa}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"a")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\aa}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\above"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0.2em"},[t("mi",null,"a"),t("mrow",null,[t("mi",null,"b"),t("mo",null,"+"),t("mn",null,"1")])])]),t("annotation",{encoding:"application/x-tex"},"{a \\above{2pt} b+1}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.4458em","vertical-align":"-0.5944em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8514em"}},[t("span",{style:{top:"-2.4639em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b"),t("span",{class:"mbin mtight"},"+"),t("span",{class:"mord mtight"},"1")])])]),t("span",{style:{top:"-3.15em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.2em"}})]),t("span",{style:{top:"-3.55em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5944em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\above{2pt} b+1}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\abovewithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\acute"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"e"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\acute e")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal"},"e")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\acute e")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AE"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"Æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AE}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"Æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AE}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ae"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\ae}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\ae}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alef"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alef")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alefsym"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alefsym")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aleph"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\aleph")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},align:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},aligned:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"b"),t("mo",null,"+"),t("mi",null,"c")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mi",null,"d"),t("mo",null,"+"),t("mi",null,"e")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"f")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{aligned}a&=b+c\\\\d+e&=f\\end{aligned}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"e")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal"},"b"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"c")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{aligned}"),t("br"),s("   "),t("code",null,"a&=b+c \\\\"),t("br"),s("   "),t("code",null,"d+e&=f"),t("br"),t("code",null,"\\end{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignat:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignedat:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"10")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"2")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"13")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"4")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{alignedat}{2}10&x+&3&y=2\\\\3&x+&13&y=4\\end{alignedat}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"10")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"13")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"2")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"4")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{alignedat}{2}"),t("br"),s("   "),t("code",null,"10&x+ &3&y = 2 \\\\"),t("br"),s("   "),t("code",null," 3&x+&13&y = 4"),t("br"),t("code",null,"\\end{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\allowbreak"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"A")]),t("annotation",{encoding:"application/x-tex"},"\\Alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"A")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"α")]),t("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\amalg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⨿")]),t("annotation",{encoding:"application/x-tex"},"\\amalg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord"},"⨿")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\And"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"&")]),t("annotation",{encoding:"application/x-tex"},"\\And")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\and"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ang"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angl"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angle"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∠")]),t("annotation",{encoding:"application/x-tex"},"\\angle")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mord"},"∠")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approx"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≈")]),t("annotation",{encoding:"application/x-tex"},"\\approx")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4831em"}}),t("span",{class:"mrel"},"≈")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approxeq"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≊")]),t("annotation",{encoding:"application/x-tex"},"\\approxeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6633em","vertical-align":"-0.0817em"}}),t("span",{class:"mrel amsrm"},"≊")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arccos"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arccos"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arccos")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mop"},"arccos")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arcct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcsin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcsin"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcsin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mop"},"arcsin")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctan"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctan"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctan")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6151em"}}),t("span",{class:"mop"},"arctan")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("ar"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmax"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg max"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmax")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"max")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg min"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"min")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},array:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\array"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arraystretch"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.66em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\def\\arraystretch{1.5}\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3.6em","vertical-align":"-1.55em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\arraystretch{1.5}"),t("br"),t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ast"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∗")]),t("annotation",{encoding:"application/x-tex"},"\\ast")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4653em"}}),t("span",{class:"mord"},"∗")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\asymp"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≍")]),t("annotation",{encoding:"application/x-tex"},"\\asymp")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4637em"}}),t("span",{class:"mrel"},"≍")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atop"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0px"},[t("mi",null,"a"),t("mi",null,"b")])]),t("annotation",{encoding:"application/x-tex"},"{a \\atop b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0904em","vertical-align":"-0.345em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7454em"}},[t("span",{style:{top:"-2.355em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.144em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\atop b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atopwithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})])])],-1),h=t("h2",{id:"b",tabindex:"-1"},[s("B "),t("a",{class:"header-anchor",href:"#b","aria-label":'Permalink to "B"'},"​")],-1),o=t("p",null,[s("| Symbol/Function | Rendered | Source or Comment | | :------------------ | :-------------------------------------------------- | :------------------------------------------------------------------------------------------------ | ------ | | \\backepsilon | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∍")]),t("annotation",{encoding:"application/x-tex"},"\\backepsilon")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mrel amsrm"},"∍")])])]),s(" | | | \\backprime | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"‵")]),t("annotation",{encoding:"application/x-tex"},"\\backprime")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5499em"}}),t("span",{class:"mord amsrm"},"‵")])])]),s(" | | | \\backsim | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∽")]),t("annotation",{encoding:"application/x-tex"},"\\backsim")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.3779em"}}),t("span",{class:"mrel amsrm"},"∽")])])]),s(" | | | \\backsimeq | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋍")]),t("annotation",{encoding:"application/x-tex"},"\\backsimeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.464em"}}),t("span",{class:"mrel amsrm"},"⋍")])])]),s(" | | | \\backslash | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"\\")]),t("annotation",{encoding:"application/x-tex"},"\\backslash")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"\\")])])]),s(" | | | \\bar | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"y"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7622em","vertical-align":"-0.1944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˉ")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.1944em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bar{y}"),s(" | | \\barwedge | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⊼")]),t("annotation",{encoding:"application/x-tex"},"\\barwedge")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8867em","vertical-align":"-0.1944em"}}),t("span",{class:"mord amsrm"},"⊼")])])]),s(" | | | \\Bbb | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"A"),t("mi",{mathvariant:"double-struck"},"B"),t("mi",{mathvariant:"double-struck"},"C")]),t("annotation",{encoding:"application/x-tex"},"\\Bbb{ABC}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbb"},"ABC")])])])]),s(" | "),t("code",null,"\\Bbb{ABC}"),t("br"),s("KaTeX supports A-Z & k | | \\Bbbk | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"k")]),t("annotation",{encoding:"application/x-tex"},"\\Bbbk")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord mathbb"},"k")])])]),s(" | | | \\bbox | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\bcancel | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("menclose",{notation:"downdiagonalstrike"},[t("mn",null,"5")])]),t("annotation",{encoding:"application/x-tex"},"\\bcancel{5}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6444em"}}),t("span",{class:"mord"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8444em"}},[t("span",{style:{top:"-3.0444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"5")])]),t("span",{class:"svg-align",style:{top:"-2.8444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{style:{height:"1.0444em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"1.0444em"},[t("line",{x1:"0",y1:"0",x2:"100%",y2:"100%","stroke-width":"0.046em"})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.2em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bcancel{5}"),s(" | | \\because | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∵")]),t("annotation",{encoding:"application/x-tex"},"\\because")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mrel amsrm"},"∵")])])]),s(" | | | \\begin | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])]),s(" | "),t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}"),s(" | | \\begingroup | "),t("span",{class:"katex-error",title:"ParseError: KaTeX parse error: Expected '\\endgroup', got '}' at position 14: \\begingroup a}̲"},"\\begingroup a}"),s(" | "),t("code",null,"\\begingroup a}"),s(" | | \\Beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"B")]),t("annotation",{encoding:"application/x-tex"},"\\Beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"B")])])]),s(" | | | \\beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"β")]),t("annotation",{encoding:"application/x-tex"},"\\beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s(" | | | \\beth | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℶ")]),t("annotation",{encoding:"application/x-tex"},"\\beth")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord amsrm"},"ℶ")])])]),s(" | | | \\between | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≬")]),t("annotation",{encoding:"application/x-tex"},"\\between")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0117em","vertical-align":"-0.2558em"}}),t("span",{class:"mrel amsrm"},"≬")])])]),s(" | | | \\bf | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"bold"},"A"),t("mi",{mathvariant:"bold"},"a"),t("mi",{mathvariant:"bold"},"B"),t("mi",{mathvariant:"bold"},"b"),t("mn",{mathvariant:"bold"},"12")]),t("annotation",{encoding:"application/x-tex"},"\\bf AaBb12")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbf"},"AaBb12")])])])]),s(" | "),t("code",null,"\\bf AaBb12"),s(" | | \\bfseries | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\big(\\big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},")")])])])]),s(" | "),t("code",null,"\\big(\\big)"),s(" | | \\Big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Big(\\Big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},")")])])])]),s(" | "),t("code",null,"\\Big(\\Big)"),s(" | | \\bigcap | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋂")]),t("annotation",{encoding:"application/x-tex"},"\\bigcap")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋂")])])]),s(" | | | \\bigcirc | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"◯")]),t("annotation",{encoding:"application/x-tex"},"\\bigcirc")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"◯")])])]),s(" | | | \\bigcup | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋃")]),t("annotation",{encoding:"application/x-tex"},"\\bigcup")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋃")])])]),s(" | | | \\bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\bigg(\\bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},")")])])])]),s(" | "),t("code",null,"\\bigg(\\bigg)"),s(" | | \\Bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Bigg(\\Bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},")")])])])]),s(" | "),t("code",null,"\\Bigg(\\Bigg)"),s(" | | \\biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size3"},"(")])])])]),s(" | "),t("code",null,"\\biggl("),s(" | | \\Biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"3em",maxsize:"3em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\Biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size4"},"(")])])])]),s(" | "),t("code",null,"\\Biggl("),s(" | | \\biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.45em"}},[t("span",{class:"pstrut",style:{height:"4.4em"}}),t("span",{style:{width:"0.333em",height:"2.400em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"2.400em",viewBox:"0 0 333 2400"},[t("path",{d:`M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15 +import{_ as e,c as a,z as t,R as n,a as s,o as l}from"./chunks/framework.a1bac067.js";const Za=JSON.parse('{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically.","frontmatter":{"title":"KaTeX: Support table","description":"Supported table of KaTeX syntax ordered alphabetically."},"headers":[],"relativePath":"application/markdown-it-katex/support-table.md","filePath":"application/markdown-it-katex/support-table.md","lastUpdated":1684859551000}'),i={name:"application/markdown-it-katex/support-table.md"},m=n("",3),c=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"n"),t("mo",{stretchy:"false"},"!")]),t("annotation",{encoding:"application/x-tex"},"n!")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"n"),t("span",{class:"mclose"},"!")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"n!")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\!"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," ⁣"),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\!b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"-0.1667em"}}),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\!b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"y"),t("mn",null,"2")])]),t("annotation",{encoding:"application/x-tex"},"\\def\\bar#1{#1^2} \\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0085em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8141em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},"2")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\bar#1{#1^2} \\bar{y}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\#"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"#")]),t("annotation",{encoding:"application/x-tex"},"\\#")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"#")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"%"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}},[t("code",null,"%this is a comment")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\%"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"%")]),t("annotation",{encoding:"application/x-tex"},"\\%")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord"},"%")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\cr c & d \\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\&"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"&")]),t("annotation",{encoding:"application/x-tex"},"\\&")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mrow"),t("mo",{mathvariant:"normal",lspace:"0em",rspace:"0em"},"′")])]),t("annotation",{encoding:"application/x-tex"},"'")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7519em"}}),t("span",{class:"mord"},[t("span"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7519em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mtight"},"′")])])])])])])])])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\'"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\'{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"("),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"(")]),t("annotation",{encoding:"application/x-tex"},"(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"(")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},")"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},")")]),t("annotation",{encoding:"application/x-tex"},")")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},")")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\(…\\)"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mfrac",null,[t("mi",null,"a"),t("mi",null,"b")])])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\(\\frac a b\\)}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0404em","vertical-align":"-0.345em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6954em"}},[t("span",{style:{top:"-2.655em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.23em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.04em"}})]),t("span",{style:{top:"-3.394em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\(\\frac a b\\)}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\ b")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace"}," "),t("span",{class:"mord mathnormal"},"b")])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\ b")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},'\\"'),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"¨")])]),t("annotation",{encoding:"application/x-tex"},'\\text{\\"{a}}')])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"¨")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,'\\text{\\"{a}}')])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\$"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"$")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\textdollar}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8056em","vertical-align":"-0.0556em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"$")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\,"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\,\\,{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\,\\,{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\."),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˙")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\.{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1389em"}},[t("span",{class:"mord"},"˙")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\.{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\:"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\:\\:{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\:\\:{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\;"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null,"  "),t("mtext",null,"  "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\;\\;{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[s("a"),t("code",null,"\\;\\;{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msub",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x_i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5806em","vertical-align":"-0.15em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.3117em"}},[t("span",{style:{top:"-2.55em","margin-left":"0em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.15em"}},[t("span")])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x_i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\_"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"_")]),t("annotation",{encoding:"application/x-tex"},"\\_")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em","vertical-align":"-0.31em"}}),t("span",{class:"mord",style:{"margin-right":"0.02778em"}},"_")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},[s("\\"),t("code",null,"`")]),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˋ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\`{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˋ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\'{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"<"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"<")]),t("annotation",{encoding:"application/x-tex"},"<")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},"<")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\="),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\={a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5678em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˉ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\={a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},">"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,">")]),t("annotation",{encoding:"application/x-tex"},">")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5782em","vertical-align":"-0.0391em"}}),t("span",{class:"mrel"},">")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\>"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a"),t("mtext",null," "),t("mtext",null," "),t("mi",null,"b")]),t("annotation",{encoding:"application/x-tex"},"a\\>\\>{b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord mathnormal"},"a"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"a\\>\\>{b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"["),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"[")]),t("annotation",{encoding:"application/x-tex"},"[")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"[")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"]"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"]")]),t("annotation",{encoding:"application/x-tex"},"]")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"]")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"a")]),t("annotation",{encoding:"application/x-tex"},"{a}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\{"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"{")]),t("annotation",{encoding:"application/x-tex"},"\\{")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mopen"},"{")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\}"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{stretchy:"false"},"}")]),t("annotation",{encoding:"application/x-tex"},"\\}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mclose"},"}")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∣")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\|"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∥")]),t("annotation",{encoding:"application/x-tex"},"\\Vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"∥")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"no no no breaks")]),t("annotation",{encoding:"application/x-tex"},"\\text{no~no~no~breaks}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"no"),t("span",{class:"mord nobreak"}," "),t("span",{class:"mord"},"breaks")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{no~no~no~breaks}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\~"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˜")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\~{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6679em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"˜")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\~{a}}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\\\"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("msup",null,[t("mi",null,"x"),t("mi",null,"i")])]),t("annotation",{encoding:"application/x-tex"},"x^i")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8247em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"x"),t("span",{class:"msupsub"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8247em"}},[t("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mathnormal mtight"},"i")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"x^i")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\^"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"ˆ")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\^{a}}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"a")])]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.25em"}},[t("span",{class:"mord"},"ˆ")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\\\^{a}}")])])])],-1),p=t("h2",{id:"a",tabindex:"-1"},[s("A "),t("a",{class:"header-anchor",href:"#a","aria-label":'Permalink to "A"'},"​")],-1),r=t("table",null,[t("thead",null,[t("tr",null,[t("th",{style:{"text-align":"left"}},"Symbol/Function"),t("th",{style:{"text-align":"left"}},"Rendered"),t("th",{style:{"text-align":"left"}},"Source or Comment")])]),t("tbody",null,[t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AA"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"A"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AA}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.9468em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.9468em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"A")]),t("span",{style:{top:"-3.2523em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AA}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aa"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mtext",null,"a"),t("mo",null,"˚")])]),t("annotation",{encoding:"application/x-tex"},"\\text{\\aa}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord text"},[t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},"a")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.375em"}},[t("span",{class:"mord"},"˚")])])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\aa}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\above"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0.2em"},[t("mi",null,"a"),t("mrow",null,[t("mi",null,"b"),t("mo",null,"+"),t("mn",null,"1")])])]),t("annotation",{encoding:"application/x-tex"},"{a \\above{2pt} b+1}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.4458em","vertical-align":"-0.5944em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8514em"}},[t("span",{style:{top:"-2.4639em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b"),t("span",{class:"mbin mtight"},"+"),t("span",{class:"mord mtight"},"1")])])]),t("span",{style:{top:"-3.15em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"frac-line",style:{"border-bottom-width":"0.2em"}})]),t("span",{style:{top:"-3.55em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5944em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\above{2pt} b+1}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\abovewithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\acute"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"e"),t("mo",null,"ˊ")])]),t("annotation",{encoding:"application/x-tex"},"\\acute e")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.6944em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal"},"e")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˊ")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\acute e")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\AE"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"Æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\AE}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"Æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\AE}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ae"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mtext",null,"æ")]),t("annotation",{encoding:"application/x-tex"},"\\text{\\ae}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord text"},[t("span",{class:"mord"},"æ")])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\text{\\ae}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alef"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alef")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alefsym"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\alefsym")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\aleph"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℵ")]),t("annotation",{encoding:"application/x-tex"},"\\aleph")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"ℵ")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},align:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},aligned:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"b"),t("mo",null,"+"),t("mi",null,"c")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mi",null,"d"),t("mo",null,"+"),t("mi",null,"e")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mo",null,"="),t("mi",null,"f")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{aligned}a&=b+c\\\\d+e&=f\\end{aligned}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"e")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal"},"b"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mbin"},"+"),t("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),t("span",{class:"mord mathnormal"},"c")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.10764em"}},"f")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{aligned}"),t("br"),s("   "),t("code",null,"a&=b+c \\\\"),t("br"),s("   "),t("code",null,"d+e&=f"),t("br"),t("code",null,"\\end{aligned}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignat:""}),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"},alignedat:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.25em",columnalign:"right left right left",columnspacing:"0em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"10")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"2")])])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"3")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"x"),t("mo",null,"+")])])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mn",null,"13")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"true"},[t("mrow",null,[t("mrow"),t("mi",null,"y"),t("mo",null,"="),t("mn",null,"4")])])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{alignedat}{2}10&x+&3&y=2\\\\3&x+&13&y=4\\end{alignedat}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"10")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal"},"x"),t("span",{class:"mord"},"+")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-r"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"3")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"13")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])]),t("span",{class:"col-align-l"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.91em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"2")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord"}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y"),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mrel"},"="),t("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),t("span",{class:"mord"},"4")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.25em"}},[t("span")])])])])])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{alignedat}{2}"),t("br"),s("   "),t("code",null,"10&x+ &3&y = 2 \\\\"),t("br"),s("   "),t("code",null," 3&x+&13&y = 4"),t("br"),t("code",null,"\\end{alignedat}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\allowbreak"),t("td",{style:{"text-align":"left"}}),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"A")]),t("annotation",{encoding:"application/x-tex"},"\\Alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"A")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\alpha"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"α")]),t("annotation",{encoding:"application/x-tex"},"\\alpha")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.0037em"}},"α")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\amalg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⨿")]),t("annotation",{encoding:"application/x-tex"},"\\amalg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord"},"⨿")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\And"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"&")]),t("annotation",{encoding:"application/x-tex"},"\\And")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},"&")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\and"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ang"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[t("a",{href:"https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax",target:"_blank",rel:"noreferrer"},"Deprecated")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angl"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\angle"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"∠")]),t("annotation",{encoding:"application/x-tex"},"\\angle")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mord"},"∠")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approx"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≈")]),t("annotation",{encoding:"application/x-tex"},"\\approx")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4831em"}}),t("span",{class:"mrel"},"≈")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\approxeq"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≊")]),t("annotation",{encoding:"application/x-tex"},"\\approxeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6633em","vertical-align":"-0.0817em"}}),t("span",{class:"mrel amsrm"},"≊")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arccos"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arccos"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arccos")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mop"},"arccos")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arcct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arcsin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arcsin"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arcsin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6679em"}}),t("span",{class:"mop"},"arcsin")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctan"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctan"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctan")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6151em"}}),t("span",{class:"mop"},"arctan")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arctg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arctg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arctg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8095em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("arct"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arg"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"arg"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\arg")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[s("ar"),t("span",{style:{"margin-right":"0.01389em"}},"g")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmax"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg max"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmax")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"max")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\argmin"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"arg min"),t("mo",null,"⁡")]),t("annotation",{encoding:"application/x-tex"},"\\argmin")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8623em","vertical-align":"-0.1944em"}}),t("span",{class:"mop"},[t("span",{class:"mord mathrm",style:{"margin-right":"0.01389em"}},"arg"),t("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),t("span",{class:"mord mathrm"},"min")])])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"},array:""}),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\array"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}},[s("see "),t("code",null,"{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arraystretch"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.66em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\def\\arraystretch{1.5}\\begin{array}{cc}a&b\\\\c&d\\end{array}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3.6em","vertical-align":"-1.55em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"2.05em"}},[t("span",{style:{top:"-4.05em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.25em"}},[t("span",{class:"pstrut",style:{height:"3.26em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.55em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"\\def\\arraystretch{1.5}"),t("br"),t("code",null,"\\begin{array}{cc}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{array}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\Arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\arrowvert"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\ast"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∗")]),t("annotation",{encoding:"application/x-tex"},"\\ast")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4653em"}}),t("span",{class:"mord"},"∗")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\asymp"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≍")]),t("annotation",{encoding:"application/x-tex"},"\\asymp")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4637em"}}),t("span",{class:"mrel"},"≍")])])])]),t("td",{style:{"text-align":"left"}})]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atop"),t("td",{style:{"text-align":"left"}},[t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mfrac",{linethickness:"0px"},[t("mi",null,"a"),t("mi",null,"b")])]),t("annotation",{encoding:"application/x-tex"},"{a \\atop b}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0904em","vertical-align":"-0.345em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},[t("span",{class:"mopen nulldelimiter"}),t("span",{class:"mfrac"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.7454em"}},[t("span",{style:{top:"-2.355em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"b")])])]),t("span",{style:{top:"-3.144em"}},[t("span",{class:"pstrut",style:{height:"2.7em"}}),t("span",{class:"sizing reset-size6 size3 mtight"},[t("span",{class:"mord mtight"},[t("span",{class:"mord mathnormal mtight"},"a")])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.345em"}},[t("span")])])])]),t("span",{class:"mclose nulldelimiter"})])])])])])]),t("td",{style:{"text-align":"left"}},[t("code",null,"{a \\atop b}")])]),t("tr",null,[t("td",{style:{"text-align":"left"}},"\\atopwithdelims"),t("td",{style:{"text-align":"left"}},[t("span",{style:{color:"firebrick"}},"Not supported")]),t("td",{style:{"text-align":"left"}})])])],-1),h=t("h2",{id:"b",tabindex:"-1"},[s("B "),t("a",{class:"header-anchor",href:"#b","aria-label":'Permalink to "B"'},"​")],-1),o=t("p",null,[s("| Symbol/Function | Rendered | Source or Comment | | :------------------ | :-------------------------------------------------- | :------------------------------------------------------------------------------------------------ | ------ | | \\backepsilon | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∍")]),t("annotation",{encoding:"application/x-tex"},"\\backepsilon")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.4306em"}}),t("span",{class:"mrel amsrm"},"∍")])])]),s(" | | | \\backprime | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"‵")]),t("annotation",{encoding:"application/x-tex"},"\\backprime")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.5499em"}}),t("span",{class:"mord amsrm"},"‵")])])]),s(" | | | \\backsim | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∽")]),t("annotation",{encoding:"application/x-tex"},"\\backsim")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.3779em"}}),t("span",{class:"mrel amsrm"},"∽")])])]),s(" | | | \\backsimeq | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋍")]),t("annotation",{encoding:"application/x-tex"},"\\backsimeq")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.464em"}}),t("span",{class:"mrel amsrm"},"⋍")])])]),s(" | | | \\backslash | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"\\")]),t("annotation",{encoding:"application/x-tex"},"\\backslash")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mord"},"\\")])])]),s(" | | | \\bar | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mover",{accent:"true"},[t("mi",null,"y"),t("mo",null,"ˉ")])]),t("annotation",{encoding:"application/x-tex"},"\\bar{y}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.7622em","vertical-align":"-0.1944em"}}),t("span",{class:"mord accent"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.5678em"}},[t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"y")]),t("span",{style:{top:"-3em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"accent-body",style:{left:"-0.1944em"}},[t("span",{class:"mord"},"ˉ")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.1944em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bar{y}"),s(" | | \\barwedge | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⊼")]),t("annotation",{encoding:"application/x-tex"},"\\barwedge")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8867em","vertical-align":"-0.1944em"}}),t("span",{class:"mord amsrm"},"⊼")])])]),s(" | | | \\Bbb | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"A"),t("mi",{mathvariant:"double-struck"},"B"),t("mi",{mathvariant:"double-struck"},"C")]),t("annotation",{encoding:"application/x-tex"},"\\Bbb{ABC}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbb"},"ABC")])])])]),s(" | "),t("code",null,"\\Bbb{ABC}"),t("br"),s("KaTeX supports A-Z & k | | \\Bbbk | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"double-struck"},"k")]),t("annotation",{encoding:"application/x-tex"},"\\Bbbk")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord mathbb"},"k")])])]),s(" | | | \\bbox | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\bcancel | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("menclose",{notation:"downdiagonalstrike"},[t("mn",null,"5")])]),t("annotation",{encoding:"application/x-tex"},"\\bcancel{5}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6444em"}}),t("span",{class:"mord"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.8444em"}},[t("span",{style:{top:"-3.0444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{class:"mord"},[t("span",{class:"mord"},"5")])]),t("span",{class:"svg-align",style:{top:"-2.8444em"}},[t("span",{class:"pstrut",style:{height:"3.0444em"}}),t("span",{style:{height:"1.0444em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"1.0444em"},[t("line",{x1:"0",y1:"0",x2:"100%",y2:"100%","stroke-width":"0.046em"})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.2em"}},[t("span")])])])])])])]),s(" | "),t("code",null,"\\bcancel{5}"),s(" | | \\because | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"∵")]),t("annotation",{encoding:"application/x-tex"},"\\because")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6922em"}}),t("span",{class:"mrel amsrm"},"∵")])])]),s(" | | | \\begin | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mtable",{rowspacing:"0.16em",columnalign:"center center",columnspacing:"1em"},[t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"a")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"b")])])]),t("mtr",null,[t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"c")])]),t("mtd",null,[t("mstyle",{scriptlevel:"0",displaystyle:"false"},[t("mi",null,"d")])])])]),t("annotation",{encoding:"application/x-tex"},"\\begin{matrix} a & b\\\\ c & d\\end{matrix}")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"mtable"},[t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"a")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"c")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])]),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"arraycolsep",style:{width:"0.5em"}}),t("span",{class:"col-align-c"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.61em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"b")])]),t("span",{style:{top:"-2.41em"}},[t("span",{class:"pstrut",style:{height:"3em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathnormal"},"d")])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])])]),s(" | "),t("code",null,"\\begin{matrix}"),t("br"),s("   "),t("code",null,"a & b \\\\"),t("br"),s("   "),t("code",null,"c & d"),t("br"),t("code",null,"\\end{matrix}"),s(" | | \\begingroup | "),t("span",{class:"katex-error",title:"ParseError: KaTeX parse error: Expected '\\endgroup', got '}' at position 14: \\begingroup a}̲"},"\\begingroup a}"),s(" | "),t("code",null,"\\begingroup a}"),s(" | | \\Beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"B")]),t("annotation",{encoding:"application/x-tex"},"\\Beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6833em"}}),t("span",{class:"mord mathrm"},"B")])])]),s(" | | | \\beta | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",null,"β")]),t("annotation",{encoding:"application/x-tex"},"\\beta")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord mathnormal",style:{"margin-right":"0.05278em"}},"β")])])]),s(" | | | \\beth | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"normal"},"ℶ")]),t("annotation",{encoding:"application/x-tex"},"\\beth")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6889em"}}),t("span",{class:"mord amsrm"},"ℶ")])])]),s(" | | | \\between | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"≬")]),t("annotation",{encoding:"application/x-tex"},"\\between")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.0117em","vertical-align":"-0.2558em"}}),t("span",{class:"mrel amsrm"},"≬")])])]),s(" | | | \\bf | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mi",{mathvariant:"bold"},"A"),t("mi",{mathvariant:"bold"},"a"),t("mi",{mathvariant:"bold"},"B"),t("mi",{mathvariant:"bold"},"b"),t("mn",{mathvariant:"bold"},"12")]),t("annotation",{encoding:"application/x-tex"},"\\bf AaBb12")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.6944em"}}),t("span",{class:"mord"},[t("span",{class:"mord mathbf"},"AaBb12")])])])]),s(" | "),t("code",null,"\\bf AaBb12"),s(" | | \\bfseries | "),t("span",{style:{color:"firebrick"}},"Not supported"),s(" | | | \\big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.2em",maxsize:"1.2em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\big(\\big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.2em","vertical-align":"-0.35em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size1"},")")])])])]),s(" | "),t("code",null,"\\big(\\big)"),s(" | | \\Big | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"1.8em",maxsize:"1.8em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Big(\\Big)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1.8em","vertical-align":"-0.65em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size2"},")")])])])]),s(" | "),t("code",null,"\\Big(\\Big)"),s(" | | \\bigcap | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋂")]),t("annotation",{encoding:"application/x-tex"},"\\bigcap")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋂")])])]),s(" | | | \\bigcirc | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"◯")]),t("annotation",{encoding:"application/x-tex"},"\\bigcirc")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"0.8889em","vertical-align":"-0.1944em"}}),t("span",{class:"mord"},"◯")])])]),s(" | | | \\bigcup | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",null,"⋃")]),t("annotation",{encoding:"application/x-tex"},"\\bigcup")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"1em","vertical-align":"-0.25em"}}),t("span",{class:"mop op-symbol small-op",style:{position:"relative",top:"0em"}},"⋃")])])]),s(" | | | \\bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\bigg(\\bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size3"},")")])])])]),s(" | "),t("code",null,"\\bigg(\\bigg)"),s(" | | \\Bigg | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"("),t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},")")]),t("annotation",{encoding:"application/x-tex"},"\\Bigg(\\Bigg)")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},"(")]),t("span",{class:"mord"},[t("span",{class:"delimsizing size4"},")")])])])]),s(" | "),t("code",null,"\\Bigg(\\Bigg)"),s(" | | \\biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size3"},"(")])])])]),s(" | "),t("code",null,"\\biggl("),s(" | | \\Biggl | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"true",stretchy:"true",minsize:"3em",maxsize:"3em"},"(")]),t("annotation",{encoding:"application/x-tex"},"\\Biggl(")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mopen"},[t("span",{class:"delimsizing size4"},"(")])])])]),s(" | "),t("code",null,"\\Biggl("),s(" | | \\biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"2.4em",maxsize:"2.4em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"2.4em","vertical-align":"-0.95em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.45em"}},[t("span",{style:{top:"-3.45em"}},[t("span",{class:"pstrut",style:{height:"4.4em"}}),t("span",{style:{width:"0.333em",height:"2.400em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"2.400em",viewBox:"0 0 333 2400"},[t("path",{d:`M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15 c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15 c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z`})])])])]),t("span",{class:"vlist-s"},"​")]),t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"0.95em"}},[t("span")])])])])])])])]),s(" | "),t("code",null,"\\biggm\\vert"),s(" | | \\Biggm | "),t("span",{class:"katex"},[t("span",{class:"katex-mathml"},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("semantics",null,[t("mrow",null,[t("mo",{fence:"false",stretchy:"true",minsize:"3em",maxsize:"3em"},"∣")]),t("annotation",{encoding:"application/x-tex"},"\\Biggm\\vert")])])]),t("span",{class:"katex-html","aria-hidden":"true"},[t("span",{class:"base"},[t("span",{class:"strut",style:{height:"3em","vertical-align":"-1.25em"}}),t("span",{class:"mrel"},[t("span",{class:"delimsizing mult"},[t("span",{class:"vlist-t vlist-t2"},[t("span",{class:"vlist-r"},[t("span",{class:"vlist",style:{height:"1.75em"}},[t("span",{style:{top:"-3.75em"}},[t("span",{class:"pstrut",style:{height:"5em"}}),t("span",{style:{width:"0.333em",height:"3.000em"}},[t("svg",{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"3.000em",viewBox:"0 0 333 3000"},[t("path",{d:`M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15 c10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15 diff --git a/assets/application_markdown-it-katex_tips.md.e64f6d16.js b/assets/application_markdown-it-katex_tips.md.2d4f69d7.js similarity index 99% rename from assets/application_markdown-it-katex_tips.md.e64f6d16.js rename to assets/application_markdown-it-katex_tips.md.2d4f69d7.js index 3be2cf1f..83f69a4a 100644 --- a/assets/application_markdown-it-katex_tips.md.e64f6d16.js +++ b/assets/application_markdown-it-katex_tips.md.2d4f69d7.js @@ -1,4 +1,4 @@ -import{_ as s,c as a,o as n,R as e}from"./chunks/framework.a1bac067.js";const l="/assets/2023-03-22-17-04-25.d7c4c068.png",o="/assets/2023-03-22-20-29-28.39883b7f.png",p="/assets/2023-05-22-10-14-11.2c283da4.png",t="/assets/2023-05-22-10-27-47.c0db4fe7.png",F=JSON.parse('{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences.","frontmatter":{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences."},"headers":[],"relativePath":"application/markdown-it-katex/tips.md","filePath":"application/markdown-it-katex/tips.md","lastUpdated":1684858945000}'),r={name:"application/markdown-it-katex/tips.md"},i=e('

Tips & Hacks ​

1: Responsive KaTeX Styling ​

1.1: Issue background ​

KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.

katex overflowing on small screen devices
◎ katex overflowing on small screen devices

This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,

katex overflowing with labels annotated
◎ katex overflowing with labels annotated

The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.

1.2: Temporary solution ​

If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.

Break/wrap formula if too wide · Issue #327 · KaTeX/KaTeX
Is there a way to break formulaundefineds if they are to wide (with respect to their parent) ?
GitHub

But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.

1.3: Finding solution ​

After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.

GitHub - vuepress-theme-hope/vuepress-theme-hope: A vuepress theme with tons of features✨
A vuepress theme with tons of features✨. Contribute to vuepress-theme-hope/vuepress-theme-hope development by creating an account on GitHub.
GitHub

Under the styles directory of the repository, several line of scss styles came across my eyes,

KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme

this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,

Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen

Consequently, the fix is easy.

1.4: Solution ​

If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.

Install global scss support using your favored package manager.

sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss

Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.

scss
...
+import{_ as s,c as a,o as n,R as e}from"./chunks/framework.a1bac067.js";const l="/assets/2023-03-22-17-04-25.d7c4c068.png",o="/assets/2023-03-22-20-29-28.39883b7f.png",p="/assets/2023-05-22-10-14-11.2c283da4.png",t="/assets/2023-05-22-10-27-47.c0db4fe7.png",F=JSON.parse('{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences.","frontmatter":{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences."},"headers":[],"relativePath":"application/markdown-it-katex/tips.md","filePath":"application/markdown-it-katex/tips.md","lastUpdated":1684859551000}'),r={name:"application/markdown-it-katex/tips.md"},i=e('
Tips & Hacks ​
1: Responsive KaTeX Styling ​
1.1: Issue background ​
KaTeX works out of the box on large screen devices such as laptops and desktop computers. But as KaTeX's built-in does not support responsiveness on it's default stylings, hence KaTeX equations might overflow out of the default width of the default application containers on small-screened mobile devices as you can see in the following image in the dev tool of chrome, when the dimension of the webpage is set to responsive or under a certain fixed device dimension, the equations rendered in KaTeX overflows out of the viewport as inspected in blue hightlight.
katex overflowing on small screen devices
◎ katex overflowing on small screen devices
This inevitably causes the viewport to break and extends the default width for KaTeX equations <div> to fit, users will be required to manually scroll right in order to view the full equation consequently as follows,
katex overflowing with labels annotated
◎ katex overflowing with labels annotated
The above situation is undoubtedly annoying for user experiences while reading documentation, consider at what scenarios users have to scroll, scroll and scroll only for viewing a single long-blocked equation, who would want to read such a text, right? But don't worry we have a simple fix for this situation via by several line of css.
1.2: Temporary solution ​
If you happened to search over KaTeX's official repo issue tracker on GitHub, there are several user-made css tweaks hack already, the fix is simple by adjusting the overflows of both x and y axes of the KaTeX render <div> blocks. The katex-display > .katex selector targets the child element of the .katex-display class that has the .katex class. This is the element that contains the KaTeX math expression. The first block of styles is mostly concerned with making sure that the KaTeX expression doesn't overflow its container and can be scrolled horizontally if needed. The second block of styles sets the font and line-height for the KaTeX expression and makes sure that its text is properly indented.
Break/wrap formula if too wide · Issue #327 · KaTeX/KaTeX
Is there a way to break formulaundefineds if they are to wide (with respect to their parent) ?
GitHub
But, the issue here is, the overflowing issue is resolved on the webpage, but the style itself left with a slight whitish "box" at the crossing corner of both horizontal as well as vertical scrollbar tracks, this might not be so explicit in the light mode of the webpage, but when it turns to dark mode, the box become annoying. Some people might say it's an easy tweak via setting the display property of the "box" element to display: none; to directly remove the box out of the page, this is a smart approach; however, while the box is gone, the two crossing tracks is going to form an untouched invisible box again against two bars without color. Thus, neither ways seems to perfectly solve the problem.
1.3: Finding solution ​
After running a quick research on Google, I found a simple hack tweak used in a theme of VuePress, vuepress-theme-hope on GitHub, the theme both integrate with KaTeX and MathJax for math supports.
GitHub - vuepress-theme-hope/vuepress-theme-hope: A vuepress theme with tons of features✨
A vuepress theme with tons of features✨. Contribute to vuepress-theme-hope/vuepress-theme-hope development by creating an account on GitHub.
GitHub
Under the styles directory of the repository, several line of scss styles came across my eyes,
KaTeX tweak fix styles in theme
◎ KaTeX tweak fix styles in theme
this scss lines only add a horizontal trackbar at the bottom of each overflowing equation while maximizing the vertical height of the equation, when user is on a small screened device as follows,
Horizontal scrollbar of equations on small screens
◎ Horizontal scrollbar of equations on small screen
Consequently, the fix is easy.
1.4: Solution ​
If you are running a documentation site like this version controlled via Node, especially VitePress without native scss supports, your fix would be installing scss support globally across the project, then add the scss styles and finally import the stylesheets to take effect globally.
Install global scss support using your favored package manager.
sh
$ npm install -D scss
$ npm install -D scss
sh
$ yarn add --dev scss
$ yarn add --dev scss
sh
$ pnpm add -D scss
$ pnpm add -D scss
Then in your scss stylesheets, add the following and link or import them to your global styles, you should be off to go with a complete fix-up for KaTeX.
scss
...
 
 // katex responsiveness fix
 .katex {
diff --git a/assets/application_markdown-it-katex_tips.md.e64f6d16.lean.js b/assets/application_markdown-it-katex_tips.md.2d4f69d7.lean.js
similarity index 92%
rename from assets/application_markdown-it-katex_tips.md.e64f6d16.lean.js
rename to assets/application_markdown-it-katex_tips.md.2d4f69d7.lean.js
index 7badb8d8..c5c94cd2 100644
--- a/assets/application_markdown-it-katex_tips.md.e64f6d16.lean.js
+++ b/assets/application_markdown-it-katex_tips.md.2d4f69d7.lean.js
@@ -1 +1 @@
-import{_ as s,c as a,o as n,R as e}from"./chunks/framework.a1bac067.js";const l="/assets/2023-03-22-17-04-25.d7c4c068.png",o="/assets/2023-03-22-20-29-28.39883b7f.png",p="/assets/2023-05-22-10-14-11.2c283da4.png",t="/assets/2023-05-22-10-27-47.c0db4fe7.png",F=JSON.parse('{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences.","frontmatter":{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences."},"headers":[],"relativePath":"application/markdown-it-katex/tips.md","filePath":"application/markdown-it-katex/tips.md","lastUpdated":1684858945000}'),r={name:"application/markdown-it-katex/tips.md"},i=e("",40),c=[i];function y(d,h,u,m,B,b){return n(),a("div",null,c)}const C=s(r,[["render",y]]);export{F as __pageData,C as default};
+import{_ as s,c as a,o as n,R as e}from"./chunks/framework.a1bac067.js";const l="/assets/2023-03-22-17-04-25.d7c4c068.png",o="/assets/2023-03-22-20-29-28.39883b7f.png",p="/assets/2023-05-22-10-14-11.2c283da4.png",t="/assets/2023-05-22-10-27-47.c0db4fe7.png",F=JSON.parse('{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences.","frontmatter":{"title":"Tips to Use markdown-it-katex","description":"Extra tips on using markdown-it-katex for the best experiences."},"headers":[],"relativePath":"application/markdown-it-katex/tips.md","filePath":"application/markdown-it-katex/tips.md","lastUpdated":1684859551000}'),r={name:"application/markdown-it-katex/tips.md"},i=e("",40),c=[i];function y(d,h,u,m,B,b){return n(),a("div",null,c)}const C=s(r,[["render",y]]);export{F as __pageData,C as default};
diff --git a/assets/development_aws_README.md.6ebeba49.js b/assets/development_aws_README.md.ea2d6a9b.js
similarity index 96%
rename from assets/development_aws_README.md.6ebeba49.js
rename to assets/development_aws_README.md.ea2d6a9b.js
index 7994d461..45f90c0c 100644
--- a/assets/development_aws_README.md.6ebeba49.js
+++ b/assets/development_aws_README.md.ea2d6a9b.js
@@ -1 +1 @@
-import{_ as e,c as a,o as s,R as o}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/README.md","filePath":"development/aws/README.md","lastUpdated":1684858945000}'),n={name:"development/aws/README.md"},t=o('
The source of the main text is here, written with asciidoctor.
If you want to build the document from the source, follow these steps.
Install ​
Prerequisites:
  • ruby
  • bundler
Install ruby libraries:
bash
bundle install --path vendor/bundle
bundle install --path vendor/bundle
Build the document ​
bash
./generate.sh main.adoc
./generate.sh main.adoc
This will generate HTML and PDF documents in the directory named dist/.
Source structure ​
  • main.adoc: Main document
  • dist/: Output document (both HTML and PDF) will be generated here
  • imgs/: Put image files here
  • fonts/, pdf-style/: Fonts and styles for PDF generation. Do not touch!
  • XXX.adoc: One section per file
',12),l=[t];function r(c,i,p,d,u,h){return s(),a("div",null,l)}const y=e(n,[["render",r]]);export{b as __pageData,y as default};
+import{_ as e,c as a,o as s,R as o}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/README.md","filePath":"development/aws/README.md","lastUpdated":1684859551000}'),n={name:"development/aws/README.md"},t=o('
The source of the main text is here, written with asciidoctor.
If you want to build the document from the source, follow these steps.
Install ​
Prerequisites:
  • ruby
  • bundler
Install ruby libraries:
bash
bundle install --path vendor/bundle
bundle install --path vendor/bundle
Build the document ​
bash
./generate.sh main.adoc
./generate.sh main.adoc
This will generate HTML and PDF documents in the directory named dist/.
Source structure ​
  • main.adoc: Main document
  • dist/: Output document (both HTML and PDF) will be generated here
  • imgs/: Put image files here
  • fonts/, pdf-style/: Fonts and styles for PDF generation. Do not touch!
  • XXX.adoc: One section per file
',12),l=[t];function r(c,i,p,d,u,h){return s(),a("div",null,l)}const y=e(n,[["render",r]]);export{b as __pageData,y as default};
diff --git a/assets/development_aws_README.md.6ebeba49.lean.js b/assets/development_aws_README.md.ea2d6a9b.lean.js
similarity index 70%
rename from assets/development_aws_README.md.6ebeba49.lean.js
rename to assets/development_aws_README.md.ea2d6a9b.lean.js
index 093ac38e..749b5feb 100644
--- a/assets/development_aws_README.md.6ebeba49.lean.js
+++ b/assets/development_aws_README.md.ea2d6a9b.lean.js
@@ -1 +1 @@
-import{_ as e,c as a,o as s,R as o}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/README.md","filePath":"development/aws/README.md","lastUpdated":1684858945000}'),n={name:"development/aws/README.md"},t=o("",12),l=[t];function r(c,i,p,d,u,h){return s(),a("div",null,l)}const y=e(n,[["render",r]]);export{b as __pageData,y as default};
+import{_ as e,c as a,o as s,R as o}from"./chunks/framework.a1bac067.js";const b=JSON.parse('{"title":"","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/README.md","filePath":"development/aws/README.md","lastUpdated":1684859551000}'),n={name:"development/aws/README.md"},t=o("",12),l=[t];function r(c,i,p,d,u,h){return s(),a("div",null,l)}const y=e(n,[["render",r]]);export{b as __pageData,y as default};
diff --git a/assets/development_aws_acknowledgement.md.d9cfe93b.js b/assets/development_aws_acknowledgement.md.2c27a486.js
similarity index 97%
rename from assets/development_aws_acknowledgement.md.d9cfe93b.js
rename to assets/development_aws_acknowledgement.md.2c27a486.js
index 30e4ea97..d51028af 100644
--- a/assets/development_aws_acknowledgement.md.d9cfe93b.js
+++ b/assets/development_aws_acknowledgement.md.2c27a486.js
@@ -1 +1 @@
-import{_ as e,c as t,o as r,R as a}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"謝辞","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/acknowledgement.md","filePath":"development/aws/acknowledgement.md","lastUpdated":1684858945000}'),o={name:"development/aws/acknowledgement.md"},n=a('
謝辞 ​
本原稿の執筆にあたり，以下の方々からの協力を得た．この場を借りて，感謝を表したい．
2021 年バージョンの contributors
  • 香取真知子氏 - ハンズオンプログラムの動作確認，文章校閲
2020 年バージョンの contributors
  • 勝俣敬寛氏 - Docker イメージの作成
  • 香取真知子氏 - ハンズオンプログラムの動作確認
  • @shuuji3 - MR !15
  • @takatama_jp - MR !14
本書の執筆には Asciidoctor を使用した．
また，本書はオープンソースの教科書として，すべての読者・ディベロッパーからのフィードバックを受け付けている． 誤植や記述の誤り，改善点など見つかったら，ぜひ Issues や Pull request を投稿していただきたい．
',8),l=[n];function s(i,p,c,_,m,h){return r(),t("div",null,l)}const u=e(o,[["render",s]]);export{g as __pageData,u as default};
+import{_ as e,c as t,o as r,R as a}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"謝辞","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/acknowledgement.md","filePath":"development/aws/acknowledgement.md","lastUpdated":1684859551000}'),o={name:"development/aws/acknowledgement.md"},n=a('
謝辞 ​
本原稿の執筆にあたり，以下の方々からの協力を得た．この場を借りて，感謝を表したい．
2021 年バージョンの contributors
  • 香取真知子氏 - ハンズオンプログラムの動作確認，文章校閲
2020 年バージョンの contributors
  • 勝俣敬寛氏 - Docker イメージの作成
  • 香取真知子氏 - ハンズオンプログラムの動作確認
  • @shuuji3 - MR !15
  • @takatama_jp - MR !14
本書の執筆には Asciidoctor を使用した．
また，本書はオープンソースの教科書として，すべての読者・ディベロッパーからのフィードバックを受け付けている． 誤植や記述の誤り，改善点など見つかったら，ぜひ Issues や Pull request を投稿していただきたい．
',8),l=[n];function s(i,p,c,_,m,h){return r(),t("div",null,l)}const u=e(o,[["render",s]]);export{g as __pageData,u as default};
diff --git a/assets/development_aws_acknowledgement.md.d9cfe93b.lean.js b/assets/development_aws_acknowledgement.md.2c27a486.lean.js
similarity index 86%
rename from assets/development_aws_acknowledgement.md.d9cfe93b.lean.js
rename to assets/development_aws_acknowledgement.md.2c27a486.lean.js
index 4b1b411d..c745991c 100644
--- a/assets/development_aws_acknowledgement.md.d9cfe93b.lean.js
+++ b/assets/development_aws_acknowledgement.md.2c27a486.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as r,R as a}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"謝辞","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/acknowledgement.md","filePath":"development/aws/acknowledgement.md","lastUpdated":1684858945000}'),o={name:"development/aws/acknowledgement.md"},n=a("",8),l=[n];function s(i,p,c,_,m,h){return r(),t("div",null,l)}const u=e(o,[["render",s]]);export{g as __pageData,u as default};
+import{_ as e,c as t,o as r,R as a}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"謝辞","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/acknowledgement.md","filePath":"development/aws/acknowledgement.md","lastUpdated":1684859551000}'),o={name:"development/aws/acknowledgement.md"},n=a("",8),l=[n];function s(i,p,c,_,m,h){return r(),t("div",null,l)}const u=e(o,[["render",s]]);export{g as __pageData,u as default};
diff --git a/assets/development_aws_appendix.md.79df306e.js b/assets/development_aws_appendix.md.7272d46f.js
similarity index 99%
rename from assets/development_aws_appendix.md.79df306e.js
rename to assets/development_aws_appendix.md.7272d46f.js
index 55ec94c8..9b6bbe79 100644
--- a/assets/development_aws_appendix.md.79df306e.js
+++ b/assets/development_aws_appendix.md.7272d46f.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as e,c as n,d as l,e as p,f as o,g as r,h as c,i as t,j as i,k as d,l as y,m as C,n as A,o as u}from"./chunks/venv_shell.dc2530bc.js";import{_ as h,c as B,o as m,R as D}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Appendix: 環境構築","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/appendix.md","filePath":"development/aws/appendix.md","lastUpdated":1684858945000}'),F={name:"development/aws/appendix.md"},g=D('
Appendix: 環境構築 ​
本書を読み進めるにあたって，ハンズオンのプログラムを実行するための環境を自分のローカルマシンにセットアップしなければならない． ここでは， AWS やコマンドラインの初心者を想定して，本章で必要なソフトウェアやライブラリのインストールなどを簡単に解説する． 以下に簡単な目次を示そう． 既に環境構築が済んでいる場合は適宜読み飛ばしていただき，関係のある箇所のみ目を通せば良い．
使用する OS は Linux/Mac/Windows のどれを用いても構わない． Windows のユーザーは， Windows Subsytem for Linux (WSL) を使用することを想定している (WSL のインストール)．
また，本書のハンズオンを実行するための Docker イメージ を提供している． これを用いると， AWS CLI/CDK や Python の設定などをスキップできるので， Docker の使用方法を知っている読者には便利だろう．
AWS アカウントの取得 ​
本書で提供するハンズオンを実際に自分で試すには，読者自身で AWS のアカウントの作成をする必要がある． 詳しいアカウントの作成の手順は 公式のドキュメンテーション に書かれているので，そちらも参照していただきたい． 以下の手順に従ってアカウントの作成を行う．
まず，ウェブブラウザから AWS コンソール にアクセスし，右上の Create an AWS Account をクリックする (figure_title で実線で囲った部分)．
サインアップ (1): AWS コンソールにアクセス
次に，遷移した先のページでメールアドレスとパスワードなどの登録を行う (figure_title)．
サインアップ (2): メールアドレス・パスワードなどの登録．
続いて，住所や電話番号などを訊かれるので，すべて入力しよう (figure_title)．
サインアップ (3): 住所・電話番号の入力
次に，クレジットカードの情報の登録を求められる (figure_title)． 個人で AWS を利用する場合は，利用料金の請求はクレジットカードを経由して行われる． クレジットカードの登録なしには AWS を使い始めることはできないことに注意．
サインアップ (4): クレジットカードの登録
次の画面では，携帯電話の SMS またはボイスメッセージを利用した本人確認が求められる (figure_title)． 希望の認証方法を選択し，自分の携帯電話番号を入力しよう．
サインアップ (5): 携帯電話による本人確認
無事に本人確認が完了すると，最後にサポートプランの選択を求められる (figure_title)． 無料の Basic support を選択しておけば問題ない．
サインアップ (6): サポートプランの選択
以上のステップにより，アカウントの作成が完了する (figure_title)． 早速ログインをして， AWS コンソールにアクセスできるか確認しておこう．
サインアップ (7): アカウントの作成が完了した
AWS のシークレットキーの作成 ​
AWS シークレットキーとは， AWS CLI や AWS CDK から AWS の API を操作するときに，ユーザー認証を行うための鍵のことである． AWS CLI/CDK を使うには，最初にシークレットキーを発行する必要がある． AWS シークレットキーの詳細は 公式ドキュメンテーション "Understanding and getting your AWS credentials" を参照．
  1. AWS コンソールにログインする．
  2. 画面右上のアカウント名をクリックし，表示されるプルダウンメニューから "My Security Credentials" を選択 (figure_title)
  3. "Access keys for CLI, SDK, & API access" の下にある "Create accesss key" のボタンをクリックする (figure_title)
  4. 表示された Access key ID, Secret access key を記録しておく (画面を閉じると二度と表示されない)．
  5. 鍵を忘れてしまった場合などは，同じ手順で再発行が可能である．
  6. 発行したシークレットキーは， ~/.aws/credentials のファイルに書き込むか，環境変数に設定するなどして使う (詳しくは AWS CLI のインストール)．
AWS シークレットキーの発行1
AWS シークレットキーの発行2
AWS Educate Starter Account を用いている場合は，次の手順でシークレットキーを確認する．
  • AWS Educate のコンソール画面から， vocareum のコンソールに移動する (figure_title)．
  • Account Details をクリックし，続いて AWS CLI: Show をクリックする．
  • aws_access_key_id, aws_secret_access_key, aws_session_token が表示される (figure_title)． ここで表示された内容を ~/.aws/credentials にコピーする (AWS CLI のインストール 参照)． aws_session_token の箇所も漏らさずコピーすること．
  • 続いて， ~/.aws/config というファイルを用意し，次の内容を書き込む． 現時点では AWS Starter Account は us-east-1 リージョンでしか利用できないためである．
toml
[default]
+import{_ as s,a,b as e,c as n,d as l,e as p,f as o,g as r,h as c,i as t,j as i,k as d,l as y,m as C,n as A,o as u}from"./chunks/venv_shell.dc2530bc.js";import{_ as h,c as B,o as m,R as D}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Appendix: 環境構築","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/appendix.md","filePath":"development/aws/appendix.md","lastUpdated":1684859551000}'),F={name:"development/aws/appendix.md"},g=D('
Appendix: 環境構築 ​
本書を読み進めるにあたって，ハンズオンのプログラムを実行するための環境を自分のローカルマシンにセットアップしなければならない． ここでは， AWS やコマンドラインの初心者を想定して，本章で必要なソフトウェアやライブラリのインストールなどを簡単に解説する． 以下に簡単な目次を示そう． 既に環境構築が済んでいる場合は適宜読み飛ばしていただき，関係のある箇所のみ目を通せば良い．
使用する OS は Linux/Mac/Windows のどれを用いても構わない． Windows のユーザーは， Windows Subsytem for Linux (WSL) を使用することを想定している (WSL のインストール)．
また，本書のハンズオンを実行するための Docker イメージ を提供している． これを用いると， AWS CLI/CDK や Python の設定などをスキップできるので， Docker の使用方法を知っている読者には便利だろう．
AWS アカウントの取得 ​
本書で提供するハンズオンを実際に自分で試すには，読者自身で AWS のアカウントの作成をする必要がある． 詳しいアカウントの作成の手順は 公式のドキュメンテーション に書かれているので，そちらも参照していただきたい． 以下の手順に従ってアカウントの作成を行う．
まず，ウェブブラウザから AWS コンソール にアクセスし，右上の Create an AWS Account をクリックする (figure_title で実線で囲った部分)．
サインアップ (1): AWS コンソールにアクセス
次に，遷移した先のページでメールアドレスとパスワードなどの登録を行う (figure_title)．
サインアップ (2): メールアドレス・パスワードなどの登録．
続いて，住所や電話番号などを訊かれるので，すべて入力しよう (figure_title)．
サインアップ (3): 住所・電話番号の入力
次に，クレジットカードの情報の登録を求められる (figure_title)． 個人で AWS を利用する場合は，利用料金の請求はクレジットカードを経由して行われる． クレジットカードの登録なしには AWS を使い始めることはできないことに注意．
サインアップ (4): クレジットカードの登録
次の画面では，携帯電話の SMS またはボイスメッセージを利用した本人確認が求められる (figure_title)． 希望の認証方法を選択し，自分の携帯電話番号を入力しよう．
サインアップ (5): 携帯電話による本人確認
無事に本人確認が完了すると，最後にサポートプランの選択を求められる (figure_title)． 無料の Basic support を選択しておけば問題ない．
サインアップ (6): サポートプランの選択
以上のステップにより，アカウントの作成が完了する (figure_title)． 早速ログインをして， AWS コンソールにアクセスできるか確認しておこう．
サインアップ (7): アカウントの作成が完了した
AWS のシークレットキーの作成 ​
AWS シークレットキーとは， AWS CLI や AWS CDK から AWS の API を操作するときに，ユーザー認証を行うための鍵のことである． AWS CLI/CDK を使うには，最初にシークレットキーを発行する必要がある． AWS シークレットキーの詳細は 公式ドキュメンテーション "Understanding and getting your AWS credentials" を参照．
  1. AWS コンソールにログインする．
  2. 画面右上のアカウント名をクリックし，表示されるプルダウンメニューから "My Security Credentials" を選択 (figure_title)
  3. "Access keys for CLI, SDK, & API access" の下にある "Create accesss key" のボタンをクリックする (figure_title)
  4. 表示された Access key ID, Secret access key を記録しておく (画面を閉じると二度と表示されない)．
  5. 鍵を忘れてしまった場合などは，同じ手順で再発行が可能である．
  6. 発行したシークレットキーは， ~/.aws/credentials のファイルに書き込むか，環境変数に設定するなどして使う (詳しくは AWS CLI のインストール)．
AWS シークレットキーの発行1
AWS シークレットキーの発行2
AWS Educate Starter Account を用いている場合は，次の手順でシークレットキーを確認する．
  • AWS Educate のコンソール画面から， vocareum のコンソールに移動する (figure_title)．
  • Account Details をクリックし，続いて AWS CLI: Show をクリックする．
  • aws_access_key_id, aws_secret_access_key, aws_session_token が表示される (figure_title)． ここで表示された内容を ~/.aws/credentials にコピーする (AWS CLI のインストール 参照)． aws_session_token の箇所も漏らさずコピーすること．
  • 続いて， ~/.aws/config というファイルを用意し，次の内容を書き込む． 現時点では AWS Starter Account は us-east-1 リージョンでしか利用できないためである．
toml
[default]
     region = us-east-1
     output = json
[default]
     region = us-east-1
diff --git a/assets/development_aws_appendix.md.79df306e.lean.js b/assets/development_aws_appendix.md.7272d46f.lean.js
similarity index 89%
rename from assets/development_aws_appendix.md.79df306e.lean.js
rename to assets/development_aws_appendix.md.7272d46f.lean.js
index 8959043f..c5dce022 100644
--- a/assets/development_aws_appendix.md.79df306e.lean.js
+++ b/assets/development_aws_appendix.md.7272d46f.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as e,c as n,d as l,e as p,f as o,g as r,h as c,i as t,j as i,k as d,l as y,m as C,n as A,o as u}from"./chunks/venv_shell.dc2530bc.js";import{_ as h,c as B,o as m,R as D}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Appendix: 環境構築","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/appendix.md","filePath":"development/aws/appendix.md","lastUpdated":1684858945000}'),F={name:"development/aws/appendix.md"},g=D("",141),b=[g];function v(k,f,_,w,E,S){return m(),B("div",null,b)}const Y=h(F,[["render",v]]);export{x as __pageData,Y as default};
+import{_ as s,a,b as e,c as n,d as l,e as p,f as o,g as r,h as c,i as t,j as i,k as d,l as y,m as C,n as A,o as u}from"./chunks/venv_shell.dc2530bc.js";import{_ as h,c as B,o as m,R as D}from"./chunks/framework.a1bac067.js";const x=JSON.parse('{"title":"Appendix: 環境構築","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/appendix.md","filePath":"development/aws/appendix.md","lastUpdated":1684859551000}'),F={name:"development/aws/appendix.md"},g=D("",141),b=[g];function v(k,f,_,w,E,S){return m(),B("div",null,b)}const Y=h(F,[["render",v]]);export{x as __pageData,Y as default};
diff --git a/assets/development_aws_assignments.md.3f42c3cd.js b/assets/development_aws_assignments.md.27524939.js
similarity index 90%
rename from assets/development_aws_assignments.md.3f42c3cd.js
rename to assets/development_aws_assignments.md.27524939.js
index ce55c9c1..7447cd0d 100644
--- a/assets/development_aws_assignments.md.3f42c3cd.js
+++ b/assets/development_aws_assignments.md.27524939.js
@@ -1 +1 @@
-import{_ as t,c as a,o as s,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"課題","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/assignments.md","filePath":"development/aws/assignments.md","lastUpdated":1684858945000}'),o={name:"development/aws/assignments.md"},r=e("h1",{id:"課題",tabindex:"-1"},[n("課題 "),e("a",{class:"header-anchor",href:"#課題","aria-label":'Permalink to "課題"'},"​")],-1),d=e("p",null,"以下の課題を提出．",-1),c=[r,d];function i(l,m,_,p,h,f){return s(),a("div",null,c)}const v=t(o,[["render",i]]);export{g as __pageData,v as default};
+import{_ as t,c as a,o as s,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"課題","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/assignments.md","filePath":"development/aws/assignments.md","lastUpdated":1684859551000}'),o={name:"development/aws/assignments.md"},r=e("h1",{id:"課題",tabindex:"-1"},[n("課題 "),e("a",{class:"header-anchor",href:"#課題","aria-label":'Permalink to "課題"'},"​")],-1),d=e("p",null,"以下の課題を提出．",-1),c=[r,d];function i(l,m,_,p,h,f){return s(),a("div",null,c)}const v=t(o,[["render",i]]);export{g as __pageData,v as default};
diff --git a/assets/development_aws_assignments.md.3f42c3cd.lean.js b/assets/development_aws_assignments.md.27524939.lean.js
similarity index 90%
rename from assets/development_aws_assignments.md.3f42c3cd.lean.js
rename to assets/development_aws_assignments.md.27524939.lean.js
index ce55c9c1..7447cd0d 100644
--- a/assets/development_aws_assignments.md.3f42c3cd.lean.js
+++ b/assets/development_aws_assignments.md.27524939.lean.js
@@ -1 +1 @@
-import{_ as t,c as a,o as s,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"課題","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/assignments.md","filePath":"development/aws/assignments.md","lastUpdated":1684858945000}'),o={name:"development/aws/assignments.md"},r=e("h1",{id:"課題",tabindex:"-1"},[n("課題 "),e("a",{class:"header-anchor",href:"#課題","aria-label":'Permalink to "課題"'},"​")],-1),d=e("p",null,"以下の課題を提出．",-1),c=[r,d];function i(l,m,_,p,h,f){return s(),a("div",null,c)}const v=t(o,[["render",i]]);export{g as __pageData,v as default};
+import{_ as t,c as a,o as s,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"課題","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/assignments.md","filePath":"development/aws/assignments.md","lastUpdated":1684859551000}'),o={name:"development/aws/assignments.md"},r=e("h1",{id:"課題",tabindex:"-1"},[n("課題 "),e("a",{class:"header-anchor",href:"#課題","aria-label":'Permalink to "課題"'},"​")],-1),d=e("p",null,"以下の課題を提出．",-1),c=[r,d];function i(l,m,_,p,h,f){return s(),a("div",null,c)}const v=t(o,[["render",i]]);export{g as __pageData,v as default};
diff --git a/assets/development_aws_author.md.09627572.js b/assets/development_aws_author.md.a50d2f4c.js
similarity index 95%
rename from assets/development_aws_author.md.09627572.js
rename to assets/development_aws_author.md.a50d2f4c.js
index 94cec1fe..485c2923 100644
--- a/assets/development_aws_author.md.09627572.js
+++ b/assets/development_aws_author.md.a50d2f4c.js
@@ -1 +1 @@
-import{_ as o,c as a,o as r,z as e,a as t}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"著者紹介","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/author.md","filePath":"development/aws/author.md","lastUpdated":1684858945000}'),n={name:"development/aws/author.md"},s=e("h1",{id:"著者紹介",tabindex:"-1"},[t("著者紹介 "),e("a",{class:"header-anchor",href:"#著者紹介","aria-label":'Permalink to "著者紹介"'},"​")],-1),l=e("p",null,"真野 智之 (Tomoyuki Mano)",-1),c=e("p",null,"情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．",-1),i=e("p",null,[t("連絡先　"),e("a",{href:"mailto:tomoyukimano@gmail.com",target:"_blank",rel:"noreferrer"},"tomoyukimano@gmail.com")],-1),m=e("p",null,[t("GitHub "),e("a",{href:"https://github.com/tomomano",target:"_blank",rel:"noreferrer"},"https://github.com/tomomano")],-1),d=[s,l,c,i,m];function _(h,p,u,f,k,b){return r(),a("div",null,d)}const x=o(n,[["render",_]]);export{v as __pageData,x as default};
+import{_ as o,c as a,o as r,z as e,a as t}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"著者紹介","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/author.md","filePath":"development/aws/author.md","lastUpdated":1684859551000}'),n={name:"development/aws/author.md"},s=e("h1",{id:"著者紹介",tabindex:"-1"},[t("著者紹介 "),e("a",{class:"header-anchor",href:"#著者紹介","aria-label":'Permalink to "著者紹介"'},"​")],-1),l=e("p",null,"真野 智之 (Tomoyuki Mano)",-1),c=e("p",null,"情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．",-1),i=e("p",null,[t("連絡先　"),e("a",{href:"mailto:tomoyukimano@gmail.com",target:"_blank",rel:"noreferrer"},"tomoyukimano@gmail.com")],-1),m=e("p",null,[t("GitHub "),e("a",{href:"https://github.com/tomomano",target:"_blank",rel:"noreferrer"},"https://github.com/tomomano")],-1),d=[s,l,c,i,m];function _(h,p,u,f,k,b){return r(),a("div",null,d)}const x=o(n,[["render",_]]);export{v as __pageData,x as default};
diff --git a/assets/development_aws_author.md.09627572.lean.js b/assets/development_aws_author.md.a50d2f4c.lean.js
similarity index 95%
rename from assets/development_aws_author.md.09627572.lean.js
rename to assets/development_aws_author.md.a50d2f4c.lean.js
index 94cec1fe..485c2923 100644
--- a/assets/development_aws_author.md.09627572.lean.js
+++ b/assets/development_aws_author.md.a50d2f4c.lean.js
@@ -1 +1 @@
-import{_ as o,c as a,o as r,z as e,a as t}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"著者紹介","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/author.md","filePath":"development/aws/author.md","lastUpdated":1684858945000}'),n={name:"development/aws/author.md"},s=e("h1",{id:"著者紹介",tabindex:"-1"},[t("著者紹介 "),e("a",{class:"header-anchor",href:"#著者紹介","aria-label":'Permalink to "著者紹介"'},"​")],-1),l=e("p",null,"真野 智之 (Tomoyuki Mano)",-1),c=e("p",null,"情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．",-1),i=e("p",null,[t("連絡先　"),e("a",{href:"mailto:tomoyukimano@gmail.com",target:"_blank",rel:"noreferrer"},"tomoyukimano@gmail.com")],-1),m=e("p",null,[t("GitHub "),e("a",{href:"https://github.com/tomomano",target:"_blank",rel:"noreferrer"},"https://github.com/tomomano")],-1),d=[s,l,c,i,m];function _(h,p,u,f,k,b){return r(),a("div",null,d)}const x=o(n,[["render",_]]);export{v as __pageData,x as default};
+import{_ as o,c as a,o as r,z as e,a as t}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"著者紹介","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/author.md","filePath":"development/aws/author.md","lastUpdated":1684859551000}'),n={name:"development/aws/author.md"},s=e("h1",{id:"著者紹介",tabindex:"-1"},[t("著者紹介 "),e("a",{class:"header-anchor",href:"#著者紹介","aria-label":'Permalink to "著者紹介"'},"​")],-1),l=e("p",null,"真野 智之 (Tomoyuki Mano)",-1),c=e("p",null,"情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．",-1),i=e("p",null,[t("連絡先　"),e("a",{href:"mailto:tomoyukimano@gmail.com",target:"_blank",rel:"noreferrer"},"tomoyukimano@gmail.com")],-1),m=e("p",null,[t("GitHub "),e("a",{href:"https://github.com/tomomano",target:"_blank",rel:"noreferrer"},"https://github.com/tomomano")],-1),d=[s,l,c,i,m];function _(h,p,u,f,k,b){return r(),a("div",null,d)}const x=o(n,[["render",_]]);export{v as __pageData,x as default};
diff --git a/assets/development_aws_aws-batch.md.83297645.js b/assets/development_aws_aws-batch.md.2225dbd4.js
similarity index 99%
rename from assets/development_aws_aws-batch.md.83297645.js
rename to assets/development_aws_aws-batch.md.2225dbd4.js
index 23088a67..d9191d13 100644
--- a/assets/development_aws_aws-batch.md.83297645.js
+++ b/assets/development_aws_aws-batch.md.2225dbd4.js
@@ -1,4 +1,4 @@
-import{_ as s,a as n,b as a,c as l,d as p,e as o,f as e,g as t,h as c,i as r,j as y,k as F,l as i,m as D,n as A,o as B,p as C,q as u,r as b,s as m}from"./chunks/cloud_development.e43e5d4a.js";import{_ as d,c as f,o as h,R as E}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-batch.md","filePath":"development/aws/aws-batch.md","lastUpdated":1684858945000}'),_={name:"development/aws/aws-batch.md"},g=E('
Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する ​
ハンズオン第三回では， ECS と Fargate を使って自動質問回答システムを構築した． シンプルながらも，複数の質問が送られた場合には並列にジョブが実行され，ユーザーに答えが返されるシステムを作ることができた． ここでは，すでに学習済みの言語モデルを用いてアプリケーションを構築した． しかし，一般的に言って，機械学習のワークフローでは自分で作ったモデルを訓練することが最初のステップにあるはずである． そこで，ハンズオン第四回では，クラウドを用いて機械学習の訓練を並列化・高速化することを考える．
本ハンズオンでは深層学習におけるハイパーパラメータ最適化を取り上げる． ハイパーパラメータとは，勾配降下法によって最適化されるニューラルネットのパラメータの外にあるパラメータのことであり，具体的にはモデルの層の幅・深さなどネットワークのアーキテクチャに関わるもの，学習率やモメンタムなどパラメータの更新則に関わるものなどが含まる． 深層学習においてハイパーパラメータの調整はとても重要なタスクである． しかしながら，ハイパーパラメータを調整するには，少しずつ条件を変えながら何度もニューラルネットを学習させる必要があり，多くの計算時間がかかる． 研究・開発においては，スループットよくたくさんのモデルの可能性を探索することが生産性を決める重要なファクターであり，ハイパーパラメータ探索を高速に解くという問題は極めて関心が高い． 本ハンズオンでは，クラウドの強力な計算リソースを利用して並列的にニューラルネットの訓練を実行することで，この問題を解く方法を学んでいこう．
Auto scaling groups (ASG) ​
ハンズオンに入っていく前に， Auto scaling groups (ASG) とよばれる EC2 の概念を知っておく必要がある．
ECS の概要を示した ??? を振り返って見てほしい． 前章 (???) でも説明したが， ECS のクラスターで計算を担う実体としては EC2 と Fargate を指定することができる． Fargate については前章で記述した． Fargate を用いると，自在にスケールする計算環境をとても簡単な設定で構築することができた． しかし， GPU を利用することができないなど，いくつかの制約があった． EC2 を使用した計算環境を指定することで，プログラミングの複雑度は増すが， GPU やその他のより高度かつ複雑な設定を伴ったクラスターを構築することができる．
EC2 クラスターには ASG と呼ばれるサービスが配置される． ASG は複数の EC2 インスタンスをロジカルな単位でグループ化することでクラスターを構成する． ASG はクラスター内に新しいインスタンスを起動する，あるいは不要になったインスタンスを停止するなどのスケーリングを担う． ASG で重要な概念として， desired capacity, minimum capacity, maximum capacity というパラメータがある． minimum capacity， maximum capacity は，それぞれクラスター内に配置できるインスタンスの数の最小値・最大値を指定するパラメータである． 前者は，クラスターに負荷がかかっていない場合でもアイドリング状態にあるインスタンスを維持することで，急に負荷が増大した時などのバッファーとして作用することができる． 後者は，負荷が急に増えたときに，過剰な数のインスタンスが起動する事態を防ぎ，経済的なコストの上限を定める役割を果たす．
desired capacity が，その時々でシステムが要求するインスタンスの数を指定する． desired capacity は，例えば 24 時間のリズムに合わせてインスタンスの数を増減させる (昼は多く夜は少なくなど) などの決まったスケジュールに基づいた設定を適用することができる． あるいはクラスター全体にかかっている負荷に応じて， desired capacity を動的に制御することも可能である． どのような基準でクラスターのスケーリングを行うかを定めるルールのことを，スケーリングポリシーとよぶ． たとえば，クラスター全体の稼働率 (負荷) を常に 80% に維持する，などのスケーリングポリシーが想定できる． この場合，クラスター全体の負荷が 80%を下回ったときにはクラスターからインスタンスが削除され，80%を超える (あるいは超えると予測される) 場合はインスタンスを追加する，という操作が ASG によって自動的に行われる．
上記のようなパラメータを検討し，ユーザーは ASG を作成する． ASG を作成したのち， ECS との連携をプログラムしてあげることで， ECS を介して ASG による EC2 クラスターにタスクを投入することが可能になる．
AWS Batch ​
AWS Batch のアイコン
先に説明したように， ECS と ASG を組み合わせることで，所望の計算クラスターを構築することが可能である． しかしながら， ECS と ASG にはかなり込み入った設定が必要であり，初心者にとっても経験者にとってもなかなか面倒なプログラミングが要求される． そこで， ECS と ASG によるクラスターの設計を自動化してくれるサービスが提供されている． それが AWS Batch である．
AWS Batch はその名のとおりバッチ (Batch) 化されたジョブ (入力データだけが異なる独立した演算が繰り返し実行されること) を想定している． 多くの科学計算や機械学習がバッチ計算に当てはまる． たとえば，初期値のパラメータを変えて複数のシミュレーションを走らせる，といったケースだ． AWS Batch を用いることの利点は，クラスターのスケーリングやジョブの割り振りはすべて自動で実行され， ユーザーはクラウドの舞台裏の詳細を気にすることなく，大量のジョブを投入できるシステムが手に入る点である． が，知識として背後では ECS/ASG/EC2 の三つ巴が協調して動作しているという点は知っておいてほしい．
AWS Batch では，ジョブの投入・管理をスムーズに行うため，次のような概念が定義されている (figure_title)． まず， ジョブ (Job) というのが，AWS Batch によって実行される一つの計算の単位である． Job definitions とはジョブの内容を定義するものであり，これには実行されるべき Docker のイメージのアドレスや，割り当てる CPU・RAM の容量，環境変数などの設定が含まれる． Job definition に基づいて個々のジョブが実行される． ジョブが実行されると，ジョブは Job queues に入る． Job queues とは，実行待ち状態にあるジョブの列のことであり，時間的に最も先頭に投入されたジョブが最初に実行される． また，複数の queue を配置し， queue ごとに priority (優先度) を設定することが可能であり， priority の高い queue に溜まったジョブが優先的に実行される (筆者はこれをディズニーランドの"ファストパス"を連想して捉えている)． Compute environment とは，先述したクラスターとほぼ同義の概念であり，計算が実行される場所 (EC2 や Fargate からなるクラスター) を指す． Compute environment には，使用する EC2 のインスタンスタイプや同時に起動するインスタンス数の上限などの簡易なスケーリングポリシーが指定されている． Job queues は Compute environment の空き状況を監視しており， それに応じてジョブを Compute environment に投下する．
以上が AWS Batch を使用するうえで理解しておかなければならない概念であるが，くどくど言葉で説明してもなかなかピンとこないだろう． ここからは，実際に自分で手を動かしながら学んでいこう．
AWS Batch の主要な概念
EC2 or Fargate?
ECS でクラスターを構成する際，計算を実行する場として EC2 と Fargate の二つの選択肢があることを説明した． それぞれ長所と短所を抱えているのだが，どのような場合にどちらを使うべきだろうか？ それを検討するため，まずは table_title を見てみよう． これは EC2 と Fargate の特徴をまとめたものである． 説明の都合上，大幅な粗視化が行われている点は留意していただきたい．
EC2 vs Fargate
EC2Fargate
Compute capacity
Medium to large
Small to medium
GPU
Yes
No
Launch speed
Slow
Fast
Task placement flexibility
Low
High
Programming complexity
High
Low
これまでに見てきたように， EC2 は最大の CPU 数・メモリーサイズが大きかったり， GPU を利用できたりするなど，単一のインスタンスでの計算能力は高い． 対して， Fargate は単一インスタンスの最大 CPU 数は 4 コアが上限である． その一方で，インスタンスの起動に要する時間は Fargate のほうが圧倒的に早く，より俊敏にクラスターのスケーリングを行うことができる． また，タスクをクラスターに投入する際のフレキシビリティも Fargate のほうが高い． フレキシビリティというのは，例えば一つのインスタンスで 2 つ以上のコンテナを走らせる，などの状況である． 単位 CPU あたりで処理されるタスクの数を最大化する際には，このような設計がしばしば採用される． プログラミングの複雑さという観点からは， Fargate のほうが一般的にシンプルな実装になる．
このように， EC2 と Fargate は互いに相補的な特性を有しており，アプリケーションによって最適な計算環境は検討される必要がある． また，EC2 と Fargate を両方用いたハイブリッドクラスターというのも定義可能であり，そのような選択肢もしばしば用いられる．
準備 ​
ハンズオンのソースコードは GitHub の handson/aws-batch にある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． また， Docker が自身のローカルマシンにインストール済みであることも必要である．
このハンズオンは， g4dn.xlarge タイプの EC2 インスタンスを使うので，アメリカ東部 (us-east-1) リージョンでは 0.526 $/hour のコストが発生する． 東京 (ap-northeast-1) を選択した場合は 0.71 $/hour のコストが発生する．
??? でも注意したが，このハンズオンを始める前に G タイプインスタンスの起動上限を AWS コンソールの EC2 管理画面から確認しよう． もし上限が 0 になっていた場合は，上限緩和の申請を行う必要がある． アプリケーションの説明 にも関連した情報を記載しているので，併せて参照されたい．
MNIST 手書き文字認識 (再訪) ​
今回のハンズオンでは，機械学習のハイパーパラメータ調整を取り上げると冒頭で述べた． その最もシンプルな例題として， ??? で扱った MNIST 手書き文字認識の問題を再度取り上げよう． ??? では，適当にチョイスしたハイパーパラメータを用いてモデルの訓練を行った． ここで使用したプログラムのハイパーパラメータとしては，確率的勾配降下法 (SGD) における学習率やモメンタムが含まれる． コードでいうと，次の行が該当する．
python
optimizer = optim.SGD(model.parameters(), lr=0.01, momentum=0.5)
optimizer = optim.SGD(model.parameters(), lr=0.01, momentum=0.5)
ここで使用された 学習率 (lr=0.01) やモメンタム (momentum=0.5) は恣意的に選択された値であり，これがベストな数値であるのかはわからない． たまたまこのチョイスが最適であるかもしれないし，もっと高い精度を出すハイパーパラメータの組が存在するかもしれない． この問題に答えるため，ハイパーパラメータサーチを行おう． 今回は，最もシンプルなアプローチとして，グリッドサーチによるハイパーパラメータサーチを行おう．
機械学習のハイパーパラメータの最適化には大きく３つのアプローチが挙げられる． グリッドサーチ，ランダムサーチ，そしてベイズ最適化による方法である．
グリッドサーチとは，ハイパーパラメータの組をある範囲の中で可能な組み合わせをすべて計算し，最適なパラメータの組を見出す方法である． 最もシンプルかつ確実な方法であるが，すべての組み合わせの可能性を愚直に計算するので計算コストが大きい．
ランダムサーチ法とは，ハイパーパラメータの組をある範囲の中でランダムに抽出し，大量に試行されたランダムな組の中から最適なパラメータの組を見出す方法である． すべての可能性を網羅的に探索できるわけではないが，調整すべきパラメータの数が多数ある場合に，グリッドサーチよりも効率的に広い探索空間をカバーすることができる．
ベイズ最適化を用いた方法では，過去の探索結果から次にどの組み合わせを探索すべきかという指標を計算し，次に探索するパラメータを決定する． これにより，理論的にはグリッドサーチやランダムサーチ法よりも少ない試行回数で最適なパラメータにたどり着くことができる．
並列化の観点でいうと，グリッドサーチとランダムサーチは各ハイパーパラメータの組の計算は独立に実行することができるため並列化が容易である． このように独立したジョブとして分割・並列化可能な問題を Embarrassingly parallel な問題とよぶ (直訳すると"恥ずかしいほど並列化可能な問題"，ということになる)． Embarrassingly parallel な問題はクラウドの強力な計算リソースを用いることで，非常なシンプルな実装で解くことができる． この章ではこのようなタイプの並列計算を取り上げる．
一方，ベイズ最適化による方法は，過去の結果をもとに次の探索が決定されるので，並列化はそれほど単純ではない． 最近では optuna などのハイパーパラメータ探索のためのライブラリが発達しており，ベイズ最適化の数理的な処理を自動で実行してくれるので便利である． これらのライブラリを使うと，もし一台のコンピュータ (ノード) の中に複数の GPU が存在する場合は，並列に計算を実行することができる． しかしながら，一台のノードにとどまらず，複数のノードをまたいだ並列化は，高度なプログラミングテクニックが必要とされるだけでなく，ノード間の接続様式などクラウドのアーキテクチャにも深く依存するものである． 本書ではここまで高度なクラウドの使用方法には立ち入らない．
まずは，本ハンズオンで使用する Docker イメージをローカルで実行してみよう．
Docker イメージのソースコードは handson/aws-batch/docker にある． 基本的に ??? のハンズオンを元にし，本ハンズオン専用の軽微な変更が施してある． 興味のある読者はソースコードも含めて読んでいただきたい．
練習として，この Docker イメージを手元でビルドするところからはじめてみよう． Dockerfile が保存されているディレクトリに移動し， mymnist という名前 (Tag) をつけてビルドを実行する．
shell
$ cd handson/aws-batch/docker
+import{_ as s,a as n,b as a,c as l,d as p,e as o,f as e,g as t,h as c,i as r,j as y,k as F,l as i,m as D,n as A,o as B,p as C,q as u,r as b,s as m}from"./chunks/cloud_development.e43e5d4a.js";import{_ as d,c as f,o as h,R as E}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-batch.md","filePath":"development/aws/aws-batch.md","lastUpdated":1684859551000}'),_={name:"development/aws/aws-batch.md"},g=E('
Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する ​
ハンズオン第三回では， ECS と Fargate を使って自動質問回答システムを構築した． シンプルながらも，複数の質問が送られた場合には並列にジョブが実行され，ユーザーに答えが返されるシステムを作ることができた． ここでは，すでに学習済みの言語モデルを用いてアプリケーションを構築した． しかし，一般的に言って，機械学習のワークフローでは自分で作ったモデルを訓練することが最初のステップにあるはずである． そこで，ハンズオン第四回では，クラウドを用いて機械学習の訓練を並列化・高速化することを考える．
本ハンズオンでは深層学習におけるハイパーパラメータ最適化を取り上げる． ハイパーパラメータとは，勾配降下法によって最適化されるニューラルネットのパラメータの外にあるパラメータのことであり，具体的にはモデルの層の幅・深さなどネットワークのアーキテクチャに関わるもの，学習率やモメンタムなどパラメータの更新則に関わるものなどが含まる． 深層学習においてハイパーパラメータの調整はとても重要なタスクである． しかしながら，ハイパーパラメータを調整するには，少しずつ条件を変えながら何度もニューラルネットを学習させる必要があり，多くの計算時間がかかる． 研究・開発においては，スループットよくたくさんのモデルの可能性を探索することが生産性を決める重要なファクターであり，ハイパーパラメータ探索を高速に解くという問題は極めて関心が高い． 本ハンズオンでは，クラウドの強力な計算リソースを利用して並列的にニューラルネットの訓練を実行することで，この問題を解く方法を学んでいこう．
Auto scaling groups (ASG) ​
ハンズオンに入っていく前に， Auto scaling groups (ASG) とよばれる EC2 の概念を知っておく必要がある．
ECS の概要を示した ??? を振り返って見てほしい． 前章 (???) でも説明したが， ECS のクラスターで計算を担う実体としては EC2 と Fargate を指定することができる． Fargate については前章で記述した． Fargate を用いると，自在にスケールする計算環境をとても簡単な設定で構築することができた． しかし， GPU を利用することができないなど，いくつかの制約があった． EC2 を使用した計算環境を指定することで，プログラミングの複雑度は増すが， GPU やその他のより高度かつ複雑な設定を伴ったクラスターを構築することができる．
EC2 クラスターには ASG と呼ばれるサービスが配置される． ASG は複数の EC2 インスタンスをロジカルな単位でグループ化することでクラスターを構成する． ASG はクラスター内に新しいインスタンスを起動する，あるいは不要になったインスタンスを停止するなどのスケーリングを担う． ASG で重要な概念として， desired capacity, minimum capacity, maximum capacity というパラメータがある． minimum capacity， maximum capacity は，それぞれクラスター内に配置できるインスタンスの数の最小値・最大値を指定するパラメータである． 前者は，クラスターに負荷がかかっていない場合でもアイドリング状態にあるインスタンスを維持することで，急に負荷が増大した時などのバッファーとして作用することができる． 後者は，負荷が急に増えたときに，過剰な数のインスタンスが起動する事態を防ぎ，経済的なコストの上限を定める役割を果たす．
desired capacity が，その時々でシステムが要求するインスタンスの数を指定する． desired capacity は，例えば 24 時間のリズムに合わせてインスタンスの数を増減させる (昼は多く夜は少なくなど) などの決まったスケジュールに基づいた設定を適用することができる． あるいはクラスター全体にかかっている負荷に応じて， desired capacity を動的に制御することも可能である． どのような基準でクラスターのスケーリングを行うかを定めるルールのことを，スケーリングポリシーとよぶ． たとえば，クラスター全体の稼働率 (負荷) を常に 80% に維持する，などのスケーリングポリシーが想定できる． この場合，クラスター全体の負荷が 80%を下回ったときにはクラスターからインスタンスが削除され，80%を超える (あるいは超えると予測される) 場合はインスタンスを追加する，という操作が ASG によって自動的に行われる．
上記のようなパラメータを検討し，ユーザーは ASG を作成する． ASG を作成したのち， ECS との連携をプログラムしてあげることで， ECS を介して ASG による EC2 クラスターにタスクを投入することが可能になる．
AWS Batch ​
AWS Batch のアイコン
先に説明したように， ECS と ASG を組み合わせることで，所望の計算クラスターを構築することが可能である． しかしながら， ECS と ASG にはかなり込み入った設定が必要であり，初心者にとっても経験者にとってもなかなか面倒なプログラミングが要求される． そこで， ECS と ASG によるクラスターの設計を自動化してくれるサービスが提供されている． それが AWS Batch である．
AWS Batch はその名のとおりバッチ (Batch) 化されたジョブ (入力データだけが異なる独立した演算が繰り返し実行されること) を想定している． 多くの科学計算や機械学習がバッチ計算に当てはまる． たとえば，初期値のパラメータを変えて複数のシミュレーションを走らせる，といったケースだ． AWS Batch を用いることの利点は，クラスターのスケーリングやジョブの割り振りはすべて自動で実行され， ユーザーはクラウドの舞台裏の詳細を気にすることなく，大量のジョブを投入できるシステムが手に入る点である． が，知識として背後では ECS/ASG/EC2 の三つ巴が協調して動作しているという点は知っておいてほしい．
AWS Batch では，ジョブの投入・管理をスムーズに行うため，次のような概念が定義されている (figure_title)． まず， ジョブ (Job) というのが，AWS Batch によって実行される一つの計算の単位である． Job definitions とはジョブの内容を定義するものであり，これには実行されるべき Docker のイメージのアドレスや，割り当てる CPU・RAM の容量，環境変数などの設定が含まれる． Job definition に基づいて個々のジョブが実行される． ジョブが実行されると，ジョブは Job queues に入る． Job queues とは，実行待ち状態にあるジョブの列のことであり，時間的に最も先頭に投入されたジョブが最初に実行される． また，複数の queue を配置し， queue ごとに priority (優先度) を設定することが可能であり， priority の高い queue に溜まったジョブが優先的に実行される (筆者はこれをディズニーランドの"ファストパス"を連想して捉えている)． Compute environment とは，先述したクラスターとほぼ同義の概念であり，計算が実行される場所 (EC2 や Fargate からなるクラスター) を指す． Compute environment には，使用する EC2 のインスタンスタイプや同時に起動するインスタンス数の上限などの簡易なスケーリングポリシーが指定されている． Job queues は Compute environment の空き状況を監視しており， それに応じてジョブを Compute environment に投下する．
以上が AWS Batch を使用するうえで理解しておかなければならない概念であるが，くどくど言葉で説明してもなかなかピンとこないだろう． ここからは，実際に自分で手を動かしながら学んでいこう．
AWS Batch の主要な概念
EC2 or Fargate?
ECS でクラスターを構成する際，計算を実行する場として EC2 と Fargate の二つの選択肢があることを説明した． それぞれ長所と短所を抱えているのだが，どのような場合にどちらを使うべきだろうか？ それを検討するため，まずは table_title を見てみよう． これは EC2 と Fargate の特徴をまとめたものである． 説明の都合上，大幅な粗視化が行われている点は留意していただきたい．
EC2 vs Fargate
EC2Fargate
Compute capacity
Medium to large
Small to medium
GPU
Yes
No
Launch speed
Slow
Fast
Task placement flexibility
Low
High
Programming complexity
High
Low
これまでに見てきたように， EC2 は最大の CPU 数・メモリーサイズが大きかったり， GPU を利用できたりするなど，単一のインスタンスでの計算能力は高い． 対して， Fargate は単一インスタンスの最大 CPU 数は 4 コアが上限である． その一方で，インスタンスの起動に要する時間は Fargate のほうが圧倒的に早く，より俊敏にクラスターのスケーリングを行うことができる． また，タスクをクラスターに投入する際のフレキシビリティも Fargate のほうが高い． フレキシビリティというのは，例えば一つのインスタンスで 2 つ以上のコンテナを走らせる，などの状況である． 単位 CPU あたりで処理されるタスクの数を最大化する際には，このような設計がしばしば採用される． プログラミングの複雑さという観点からは， Fargate のほうが一般的にシンプルな実装になる．
このように， EC2 と Fargate は互いに相補的な特性を有しており，アプリケーションによって最適な計算環境は検討される必要がある． また，EC2 と Fargate を両方用いたハイブリッドクラスターというのも定義可能であり，そのような選択肢もしばしば用いられる．
準備 ​
ハンズオンのソースコードは GitHub の handson/aws-batch にある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． また， Docker が自身のローカルマシンにインストール済みであることも必要である．
このハンズオンは， g4dn.xlarge タイプの EC2 インスタンスを使うので，アメリカ東部 (us-east-1) リージョンでは 0.526 $/hour のコストが発生する． 東京 (ap-northeast-1) を選択した場合は 0.71 $/hour のコストが発生する．
??? でも注意したが，このハンズオンを始める前に G タイプインスタンスの起動上限を AWS コンソールの EC2 管理画面から確認しよう． もし上限が 0 になっていた場合は，上限緩和の申請を行う必要がある． アプリケーションの説明 にも関連した情報を記載しているので，併せて参照されたい．
MNIST 手書き文字認識 (再訪) ​
今回のハンズオンでは，機械学習のハイパーパラメータ調整を取り上げると冒頭で述べた． その最もシンプルな例題として， ??? で扱った MNIST 手書き文字認識の問題を再度取り上げよう． ??? では，適当にチョイスしたハイパーパラメータを用いてモデルの訓練を行った． ここで使用したプログラムのハイパーパラメータとしては，確率的勾配降下法 (SGD) における学習率やモメンタムが含まれる． コードでいうと，次の行が該当する．
python
optimizer = optim.SGD(model.parameters(), lr=0.01, momentum=0.5)
optimizer = optim.SGD(model.parameters(), lr=0.01, momentum=0.5)
ここで使用された 学習率 (lr=0.01) やモメンタム (momentum=0.5) は恣意的に選択された値であり，これがベストな数値であるのかはわからない． たまたまこのチョイスが最適であるかもしれないし，もっと高い精度を出すハイパーパラメータの組が存在するかもしれない． この問題に答えるため，ハイパーパラメータサーチを行おう． 今回は，最もシンプルなアプローチとして，グリッドサーチによるハイパーパラメータサーチを行おう．
機械学習のハイパーパラメータの最適化には大きく３つのアプローチが挙げられる． グリッドサーチ，ランダムサーチ，そしてベイズ最適化による方法である．
グリッドサーチとは，ハイパーパラメータの組をある範囲の中で可能な組み合わせをすべて計算し，最適なパラメータの組を見出す方法である． 最もシンプルかつ確実な方法であるが，すべての組み合わせの可能性を愚直に計算するので計算コストが大きい．
ランダムサーチ法とは，ハイパーパラメータの組をある範囲の中でランダムに抽出し，大量に試行されたランダムな組の中から最適なパラメータの組を見出す方法である． すべての可能性を網羅的に探索できるわけではないが，調整すべきパラメータの数が多数ある場合に，グリッドサーチよりも効率的に広い探索空間をカバーすることができる．
ベイズ最適化を用いた方法では，過去の探索結果から次にどの組み合わせを探索すべきかという指標を計算し，次に探索するパラメータを決定する． これにより，理論的にはグリッドサーチやランダムサーチ法よりも少ない試行回数で最適なパラメータにたどり着くことができる．
並列化の観点でいうと，グリッドサーチとランダムサーチは各ハイパーパラメータの組の計算は独立に実行することができるため並列化が容易である． このように独立したジョブとして分割・並列化可能な問題を Embarrassingly parallel な問題とよぶ (直訳すると"恥ずかしいほど並列化可能な問題"，ということになる)． Embarrassingly parallel な問題はクラウドの強力な計算リソースを用いることで，非常なシンプルな実装で解くことができる． この章ではこのようなタイプの並列計算を取り上げる．
一方，ベイズ最適化による方法は，過去の結果をもとに次の探索が決定されるので，並列化はそれほど単純ではない． 最近では optuna などのハイパーパラメータ探索のためのライブラリが発達しており，ベイズ最適化の数理的な処理を自動で実行してくれるので便利である． これらのライブラリを使うと，もし一台のコンピュータ (ノード) の中に複数の GPU が存在する場合は，並列に計算を実行することができる． しかしながら，一台のノードにとどまらず，複数のノードをまたいだ並列化は，高度なプログラミングテクニックが必要とされるだけでなく，ノード間の接続様式などクラウドのアーキテクチャにも深く依存するものである． 本書ではここまで高度なクラウドの使用方法には立ち入らない．
まずは，本ハンズオンで使用する Docker イメージをローカルで実行してみよう．
Docker イメージのソースコードは handson/aws-batch/docker にある． 基本的に ??? のハンズオンを元にし，本ハンズオン専用の軽微な変更が施してある． 興味のある読者はソースコードも含めて読んでいただきたい．
練習として，この Docker イメージを手元でビルドするところからはじめてみよう． Dockerfile が保存されているディレクトリに移動し， mymnist という名前 (Tag) をつけてビルドを実行する．
shell
$ cd handson/aws-batch/docker
 $ docker build -t mymnist .
$ cd handson/aws-batch/docker
 $ docker build -t mymnist .
docker build でエラーが出たときは次の可能性を疑ってほしい． ビルドの中で， MNIST の画像データセットを http://yann.lecun.com/exdb/mnist/ からダウンロードするのだが，ダウンロード先のサーバーがしばしばダウンしている． 世界中の機械学習ユーザーがアクセスするので，これはしばしば発生するようである． サーバーがダウンしているとビルドも失敗してしまう． エラーメッセージにそれらしい文言が含まれていたら，この可能性を疑おう．
手元でビルドするかわりに， Docker Hub から pull することも可能である． その場合は次のコマンドを実行する．
shell
$ docker pull tomomano/mymnist:latest
$ docker pull tomomano/mymnist:latest
イメージの準備ができたら，次のコマンドでコンテナを起動し， MNIST の学習を実行する．．
shell
$ docker run mymnist --lr 0.1 --momentum 0.5 --epochs 10
$ docker run mymnist --lr 0.1 --momentum 0.5 --epochs 10
このコマンドを実行すると，指定したハイパーパラメータ (--lr で与えられる学習率と --momentum で与えられるモメンタム) を使ってニューラルネットの最適化が始まる． 学習を行う最大のエポック数は --epochs パラメータで指定する． ??? のハンズオンで見たような， Loss の低下がコマンドライン上に出力されるだろう (figure_title)．
Docker を実行した際の出力
上に示したコマンドを使うと，計算は CPU を使って実行される． もし，ローカルの計算機に GPU が備わっており， nvidia-docker の設定が済んでいるならば， 次のコマンドにより GPU を使って計算を実行できる．
$ docker run --gpus all mymnist --lr 0.1 --momentum 0.5 --epochs 10
 
このコマンドでは，--gpus all というパラメータが加わった．
CPU/GPU どちらで実行した場合でも，エポックを重ねるにつれて訓練データ (Train データ) の Loss は単調に減少していくのが見て取れるだろう． 一方，検証データ (Validation データ) の Loss および Accuracy は，ある程度まで減少した後，それ以上性能が向上しないことに気がつくだろう． これを実際にプロットしてみると figure_title のようになるはずである．
(左) Train/Validation データそれぞれの Loss のエポックごとの変化． (右) Validation データの Accuracy のエポックごとの変化
これはオーバーフィッティングとよばれる現象で，ニューラルネットが訓練データに過度に最適化され，訓練データの外のデータに対しての精度 (汎化性能) が向上していないことを示している． このような場合の対処法として， Early stopping とよばれるテクニックが知られている． Early stopping とは，検証データの Loss を追跡し，それが減少から増加に転じるエポックで学習をうち止め，そのエポックでのウェイトパラメータを採用する，というものである． 本ハンズオンでも， Early stopping によって訓練の終了を判断し，モデルの性能評価を行っていく．
MNIST 手書き文字データセットでは，訓練データとして 60,000 枚，テストデータとして 10,000 枚の画像が与えられている． 本ハンズオンで使用するコードでは，訓練データのうち 80% の 48,000 枚を訓練データとして使用し，残り 20% の 12,000 枚を検証データとして用いている． 詳しくはソースコードを参照のこと．
アプリケーションの説明 ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
アプリケーションのアーキテクチャ
簡単にまとめると，次のような設計である．
  • クライアントは，あるハイパーパラメータの組を指定して Batch にジョブを提出する
  • Batch はジョブを受け取ると， EC2 からなるクラスターで計算を実行する
  • クラスター内では g4dn.xlarge インスタンスが起動する
  • Docker イメージは， AWS 内に用意された ECR (Elastic Container Registry) から取得される
  • 複数のジョブが投下された場合は，その数だけのインスタンスが起動し並列に実行される
  • 各ジョブによる計算の結果は S3 に保存される
  • 最後にクライアントは S3 から結果をダウンロードし，最適なハイパーパラメータの組を決定する
それでは，プログラムのソースコードを見てみよう (handson/aws-batch/app.py)．
python
class SimpleBatch(core.Stack):
diff --git a/assets/development_aws_aws-batch.md.83297645.lean.js b/assets/development_aws_aws-batch.md.2225dbd4.lean.js
similarity index 91%
rename from assets/development_aws_aws-batch.md.83297645.lean.js
rename to assets/development_aws_aws-batch.md.2225dbd4.lean.js
index 73d15868..0d2a2669 100644
--- a/assets/development_aws_aws-batch.md.83297645.lean.js
+++ b/assets/development_aws_aws-batch.md.2225dbd4.lean.js
@@ -1 +1 @@
-import{_ as s,a as n,b as a,c as l,d as p,e as o,f as e,g as t,h as c,i as r,j as y,k as F,l as i,m as D,n as A,o as B,p as C,q as u,r as b,s as m}from"./chunks/cloud_development.e43e5d4a.js";import{_ as d,c as f,o as h,R as E}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-batch.md","filePath":"development/aws/aws-batch.md","lastUpdated":1684858945000}'),_={name:"development/aws/aws-batch.md"},g=E("",163),q=[g];function v(k,S,j,x,w,N){return h(),f("div",null,q)}const U=d(_,[["render",v]]);export{T as __pageData,U as default};
+import{_ as s,a as n,b as a,c as l,d as p,e as o,f as e,g as t,h as c,i as r,j as y,k as F,l as i,m as D,n as A,o as B,p as C,q as u,r as b,s as m}from"./chunks/cloud_development.e43e5d4a.js";import{_ as d,c as f,o as h,R as E}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-batch.md","filePath":"development/aws/aws-batch.md","lastUpdated":1684859551000}'),_={name:"development/aws/aws-batch.md"},g=E("",163),q=[g];function v(k,S,j,x,w,N){return h(),f("div",null,q)}const U=d(_,[["render",v]]);export{T as __pageData,U as default};
diff --git a/assets/development_aws_aws-get-started.md.df995644.js b/assets/development_aws_aws-get-started.md.305acfd1.js
similarity index 99%
rename from assets/development_aws_aws-get-started.md.df995644.js
rename to assets/development_aws_aws-get-started.md.305acfd1.js
index 9526bcf4..a01aed70 100644
--- a/assets/development_aws_aws-get-started.md.df995644.js
+++ b/assets/development_aws_aws-get-started.md.305acfd1.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as o,e as p,f as e,g as t,h as r,i as c,j as y}from"./chunks/iac.2263bc12.js";import{_ as F,a as D}from"./chunks/VPC.e1acca4d.js";import{_ as A,c as i,o as C,R as B}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"AWS 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-get-started.md","filePath":"development/aws/aws-get-started.md","lastUpdated":1684858945000}'),u={name:"development/aws/aws-get-started.md"},d=B('
AWS 入門 ​
AWS とは？ ​
本書では，クラウドの実践を行うプラットフォームとして， AWS を用いる． 実践にあたって，最低限必要な AWS の知識を本章では解説しよう．
AWS (Amazon Web Services) は Amazon 社が提供する総合的なクラウドプラットフォームである． AWS は Amazon 社が持つ膨大な計算リソースを貸し出すクラウドサービスとして，2006 年に誕生した． 2021 年では，クラウドプロバイダーとして最大のマーケットシェア (約 32%) を保持している (参照)． Netflix や Slack をはじめとした多くのウェブ関連のサービスで，一部または全てのサーバーリソースが AWS から提供されているとのことである． よって，知らないうちに AWS の恩恵にあずかっている人も少なくないはずだ．
最大のシェアをもつだけに，機能・サービスの幅広さはほかのクラウドプラットフォームと比べ抜きんでている． また，利用者数が多いことを反映して，公式あるいはサードパーティによる技術紹介記事が数多くウェブ上に存在しているだけでなく，ライブラリのユーザーコミュニティも大きく問題解決が捗るのも魅力の一つだ． 初期のころウェブビジネスを行う企業がユーザーの大半を占めていたが，最近は大学などでの科学研究用途としても頻繁に用いられるようになってきている．
AWS の機能・サービス ​
figure_title は，執筆時点において AWS で提供されている主要な機能・サービスの一覧である．
AWSで提供されている主要なサービス一覧
計算，ストレージ，データベース，ネットワーク，セキュリティなど，クラウドの構築に必要な様々な要素が独立したコンポーネントとして提供されている． 基本的に，これらを組み合わせることで一つのクラウドシステムができあがる．
また，機械学習・音声認識・AR/VR など，特定のアプリケーションにパッケージ済みのサービスも提供されている． これらを合計すると全部で 170 個以上のサービスが提供されているとのことである (参照)．
AWS の初心者が陥りがちなのは，大量のサービスの数に圧倒され，どこから手をつけたらよいのかわからなくなる，という状況である． たくさんのサービスの中から，どのサービスをどの順番で学んでいったらいいのか，その道筋すら明らかでなく，大きな参入障壁となっていることは間違いない． だが実のところ， AWS の基本的な構成要素はそのうちの数個のみに限られる． 基本要素となる機能の使い方を知れば， AWS のおおよそのリソースを使いこなすことが可能になる． ほかの機能の多くは，基本の要素を組み合わせて特定のアプリケーションに特化したパッケージとして AWS が用意したものである． そのポイントを認知することが， AWS の学習の最初のステップである．
ここでは， AWS 上でクラウドシステムを構築するときの基本となる構成要素を列挙する． これらは後のハンズオンで実際にプログラムを書きながら体験する． 現時点では，名前だけでも頭の片隅に記憶してもらえればよい．
計算 ​
S3 EC2 (Elastic Compute Cloud) 様々なスペックの仮想マシンを作成し，計算を実行することができる． クラウドの最も基本となる構成要素である． ???, ???, ??? で詳しく触れる．
S3 Lambda Function as a Service (FaaS) とよばれる，小さな計算をサーバーなしで実行するためのサービス． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
ストレージ ​
S3 EBS (Elastic Block Store) EC2 に付与することのできる仮想データドライブ． いわゆる"普通の"(一般的な OS で使われている)ファイルシステムを思い浮かべてくれたらよい．
S3 S3 (Simple Storage Service) Object Storage とよばれる，API を使ってデータの読み書きを行う，いうなれば”クラウド・ネイティブ”なデータの格納システムである． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
データベース ​
S3 DynamoDB NoSQL 型のデータベースサービス (知っている人は mongoDB などを思い浮かべたらよい)． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
ネットワーク ​
S3 VPC(Virtual Private Cloud) AWS 上に仮想ネットワーク環境を作成し，仮想サーバー間の接続を定義したり，外部からのアクセスなどを管理する． EC2 は VPC の内部に配置されなければならない．
API Gateway S3 API のエンドポイントとバックエンドのサービス (Lambda など) を接続する際に用いる，リバースプロキシとしての役割を担う． ??? で詳しく解説する．
Region と Availability Zone ​
AWS を使用する際に知っておかなければならない重要な概念として， リージョン (Region) と Availability Zone (AZ) がある (figure_title)． 以下ではこの概念について簡単に記述する．
AWSにおける Region と Availability Zones
リージョン (Region) とは，おおまかに言うとデータセンターの所在地のことである． 執筆時点において， AWS は世界の 25 の国と地域でデータセンターを所有している． figure_title は執筆時点で利用できるリージョンの世界地図を示している． 日本では東京と大阪にデータセンターがある． 各リージョンには固有の ID がついており，例えば東京は ap-northeast-1, 米国オハイオ州は us-east-2，などと定義されている．
Regions in AWS(出典: https://aws.amazon.com/about-aws/global-infrastructure/)
AWS コンソールにログインすると，画面右上のメニューバーでリージョンを選択することができる(figure_title, 赤丸で囲った箇所)． EC2, S3 などの AWS のリソースは，リージョンごとに完全に独立である． したがって，リソースを新たにデプロイする際，あるいはデプロイ済みのリソースを閲覧する際は，コンソールのリージョンが正しく設定されているか，確認する必要がある． ウェブビジネスを展開する場合などは，世界の各地にクラウドを展開する必要があるが，個人的な研究用途として用いる場合は，最寄りのリージョン (i.e. 東京) を使えば基本的に問題ない．
AWSコンソールでリージョンを選択
Avaialibity Zone (AZ) とは，リージョン内で地理的に隔離されたデータセンターのことである． それぞれのリージョンは 2 個以上の AZ を有しており，もし一つの AZ で火災や停電などが起きた場合でも，ほかの AZ がその障害をカバーすることができる． また， AZ 間は高速な AWS 専用ネットワーク回線で結ばれているため， AZ 間のデータ転送は極めて早い． AZ は，ビジネスなどでサーバーダウンが許容されない場合などに注意すべき概念であり，個人的な用途で使う限りにおいてはあまり深く考慮する必要はない．言葉の意味だけ知っておけば十分である．
AWS を使用する際，どこのリージョンを指定するのがよいのだろうか？ インターネットの接続速度の観点からは，地理的に一番近いリージョンを使用するのが一般的によいだろう． 一方， EC2 の利用料などはリージョンごとに価格設定が若干 (10-20%程度) 異なる． したがって，自分が最も頻繁に利用するサービスの価格が最も安く設定されているリージョンを選択する，というのも重要な視点である． また，いくつかのサービスは，特定のリージョンで利用できない場合もある． これらのポイントから総合的に判断して使用するリージョンを決めると良い．
AWS Educate を利用している読者へ
執筆時点において，AWS Educate による Starter Account を使用している場合は us-east-1 region のみ利用できる (参照)．
AWS でのクラウド開発 ​
AWS のクラウドの全体像がわかってきたところで，次のトピックとして，どのようにして AWS 上にクラウドの開発を行い，展開していくかについての概略を解説しよう．
AWS のリソースを追加・編集・削除するなどの操作を実行するには，コンソールを用いる方法と，API を用いる方法の，二つの経路がある．
コンソール画面からリソースを操作する ​
AWS のアカウントにログインすると，まず最初に表示されるのがAWS コンソールである (figure_title)．
AWSマネージメントコンソール画面
コンソールを使うことで， EC2 のインスタンスを立ち上げたり，S3 のデータを追加・削除したり，ログを閲覧したりなど，AWS 上のあらゆるリソースの操作を GUI (Graphical User Interface) を通して実行することができる． 初めて触る機能をポチポチと試したり，デバッグを行うときなどにとても便利である．
コンソールはさらっと機能を試したり，開発中のクラウドのデバッグをするときには便利なのであるが，実際にクラウドの開発をする場面でこれを直接いじることはあまりない． むしろ，次に紹介する API を使用して，プログラムとしてクラウドのリソースを記述することで開発を行うのが一般的である． そのような理由で，本書では AWS コンソールを使った AWS の使い方はあまり触れない． AWS のドキュメンテーションには，たくさんの チュートリアル が用意されており，コンソール画面から様々な操作を行う方法が記述されているので，興味がある読者はそちらを参照されたい．
API からリソースを操作する ​
API (Application Programming Interface) を使うことで，コマンドを AWS に送信し，クラウドのリソースの操作をすることができる． API とは，端的に言えば AWS が公開しているコマンドの一覧であり，GET, POST, DELETE などの REST API から構成されている (REST API については ??? で簡単に解説する)． が，直接 REST API を入力するのは面倒であるので，その手間を解消するための様々なツールが提供されている．
例えば， AWS CLI は， UNIX コンソールから AWS API を実行するための CLI (Command Line Interface) である． CLI に加えて，いろいろなプログラミング言語での SDK (Software Development Kit) が提供されている．以下に一例を挙げる．
具体的な API の使用例を見てみよう．
S3 に新しい保存領域 (Bucket (バケット) とよばれる) を追加したいとしよう． AWS CLI を使った場合は，次のようなコマンドを打てばよい．
shell
$ aws s3 mb s3://my-bucket --region ap-northeast-1
$ aws s3 mb s3://my-bucket --region ap-northeast-1
上記のコマンドは， my-bucket という名前のバケットを， ap-northeast-1 のリージョンに作成する．
Python からこれと同じ操作を実行するには， boto3 ライブラリを使って，次のようなスクリプトを実行する．
python
import boto3
+import{_ as s,a,b as n,c as l,d as o,e as p,f as e,g as t,h as r,i as c,j as y}from"./chunks/iac.2263bc12.js";import{_ as F,a as D}from"./chunks/VPC.e1acca4d.js";import{_ as A,c as i,o as C,R as B}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"AWS 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-get-started.md","filePath":"development/aws/aws-get-started.md","lastUpdated":1684859551000}'),u={name:"development/aws/aws-get-started.md"},d=B('
AWS 入門 ​
AWS とは？ ​
本書では，クラウドの実践を行うプラットフォームとして， AWS を用いる． 実践にあたって，最低限必要な AWS の知識を本章では解説しよう．
AWS (Amazon Web Services) は Amazon 社が提供する総合的なクラウドプラットフォームである． AWS は Amazon 社が持つ膨大な計算リソースを貸し出すクラウドサービスとして，2006 年に誕生した． 2021 年では，クラウドプロバイダーとして最大のマーケットシェア (約 32%) を保持している (参照)． Netflix や Slack をはじめとした多くのウェブ関連のサービスで，一部または全てのサーバーリソースが AWS から提供されているとのことである． よって，知らないうちに AWS の恩恵にあずかっている人も少なくないはずだ．
最大のシェアをもつだけに，機能・サービスの幅広さはほかのクラウドプラットフォームと比べ抜きんでている． また，利用者数が多いことを反映して，公式あるいはサードパーティによる技術紹介記事が数多くウェブ上に存在しているだけでなく，ライブラリのユーザーコミュニティも大きく問題解決が捗るのも魅力の一つだ． 初期のころウェブビジネスを行う企業がユーザーの大半を占めていたが，最近は大学などでの科学研究用途としても頻繁に用いられるようになってきている．
AWS の機能・サービス ​
figure_title は，執筆時点において AWS で提供されている主要な機能・サービスの一覧である．
AWSで提供されている主要なサービス一覧
計算，ストレージ，データベース，ネットワーク，セキュリティなど，クラウドの構築に必要な様々な要素が独立したコンポーネントとして提供されている． 基本的に，これらを組み合わせることで一つのクラウドシステムができあがる．
また，機械学習・音声認識・AR/VR など，特定のアプリケーションにパッケージ済みのサービスも提供されている． これらを合計すると全部で 170 個以上のサービスが提供されているとのことである (参照)．
AWS の初心者が陥りがちなのは，大量のサービスの数に圧倒され，どこから手をつけたらよいのかわからなくなる，という状況である． たくさんのサービスの中から，どのサービスをどの順番で学んでいったらいいのか，その道筋すら明らかでなく，大きな参入障壁となっていることは間違いない． だが実のところ， AWS の基本的な構成要素はそのうちの数個のみに限られる． 基本要素となる機能の使い方を知れば， AWS のおおよそのリソースを使いこなすことが可能になる． ほかの機能の多くは，基本の要素を組み合わせて特定のアプリケーションに特化したパッケージとして AWS が用意したものである． そのポイントを認知することが， AWS の学習の最初のステップである．
ここでは， AWS 上でクラウドシステムを構築するときの基本となる構成要素を列挙する． これらは後のハンズオンで実際にプログラムを書きながら体験する． 現時点では，名前だけでも頭の片隅に記憶してもらえればよい．
計算 ​
S3 EC2 (Elastic Compute Cloud) 様々なスペックの仮想マシンを作成し，計算を実行することができる． クラウドの最も基本となる構成要素である． ???, ???, ??? で詳しく触れる．
S3 Lambda Function as a Service (FaaS) とよばれる，小さな計算をサーバーなしで実行するためのサービス． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
ストレージ ​
S3 EBS (Elastic Block Store) EC2 に付与することのできる仮想データドライブ． いわゆる"普通の"(一般的な OS で使われている)ファイルシステムを思い浮かべてくれたらよい．
S3 S3 (Simple Storage Service) Object Storage とよばれる，API を使ってデータの読み書きを行う，いうなれば”クラウド・ネイティブ”なデータの格納システムである． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
データベース ​
S3 DynamoDB NoSQL 型のデータベースサービス (知っている人は mongoDB などを思い浮かべたらよい)． サーバーレスアーキテクチャの章 (???) で詳しく解説する．
ネットワーク ​
S3 VPC(Virtual Private Cloud) AWS 上に仮想ネットワーク環境を作成し，仮想サーバー間の接続を定義したり，外部からのアクセスなどを管理する． EC2 は VPC の内部に配置されなければならない．
API Gateway S3 API のエンドポイントとバックエンドのサービス (Lambda など) を接続する際に用いる，リバースプロキシとしての役割を担う． ??? で詳しく解説する．
Region と Availability Zone ​
AWS を使用する際に知っておかなければならない重要な概念として， リージョン (Region) と Availability Zone (AZ) がある (figure_title)． 以下ではこの概念について簡単に記述する．
AWSにおける Region と Availability Zones
リージョン (Region) とは，おおまかに言うとデータセンターの所在地のことである． 執筆時点において， AWS は世界の 25 の国と地域でデータセンターを所有している． figure_title は執筆時点で利用できるリージョンの世界地図を示している． 日本では東京と大阪にデータセンターがある． 各リージョンには固有の ID がついており，例えば東京は ap-northeast-1, 米国オハイオ州は us-east-2，などと定義されている．
Regions in AWS(出典: https://aws.amazon.com/about-aws/global-infrastructure/)
AWS コンソールにログインすると，画面右上のメニューバーでリージョンを選択することができる(figure_title, 赤丸で囲った箇所)． EC2, S3 などの AWS のリソースは，リージョンごとに完全に独立である． したがって，リソースを新たにデプロイする際，あるいはデプロイ済みのリソースを閲覧する際は，コンソールのリージョンが正しく設定されているか，確認する必要がある． ウェブビジネスを展開する場合などは，世界の各地にクラウドを展開する必要があるが，個人的な研究用途として用いる場合は，最寄りのリージョン (i.e. 東京) を使えば基本的に問題ない．
AWSコンソールでリージョンを選択
Avaialibity Zone (AZ) とは，リージョン内で地理的に隔離されたデータセンターのことである． それぞれのリージョンは 2 個以上の AZ を有しており，もし一つの AZ で火災や停電などが起きた場合でも，ほかの AZ がその障害をカバーすることができる． また， AZ 間は高速な AWS 専用ネットワーク回線で結ばれているため， AZ 間のデータ転送は極めて早い． AZ は，ビジネスなどでサーバーダウンが許容されない場合などに注意すべき概念であり，個人的な用途で使う限りにおいてはあまり深く考慮する必要はない．言葉の意味だけ知っておけば十分である．
AWS を使用する際，どこのリージョンを指定するのがよいのだろうか？ インターネットの接続速度の観点からは，地理的に一番近いリージョンを使用するのが一般的によいだろう． 一方， EC2 の利用料などはリージョンごとに価格設定が若干 (10-20%程度) 異なる． したがって，自分が最も頻繁に利用するサービスの価格が最も安く設定されているリージョンを選択する，というのも重要な視点である． また，いくつかのサービスは，特定のリージョンで利用できない場合もある． これらのポイントから総合的に判断して使用するリージョンを決めると良い．
AWS Educate を利用している読者へ
執筆時点において，AWS Educate による Starter Account を使用している場合は us-east-1 region のみ利用できる (参照)．
AWS でのクラウド開発 ​
AWS のクラウドの全体像がわかってきたところで，次のトピックとして，どのようにして AWS 上にクラウドの開発を行い，展開していくかについての概略を解説しよう．
AWS のリソースを追加・編集・削除するなどの操作を実行するには，コンソールを用いる方法と，API を用いる方法の，二つの経路がある．
コンソール画面からリソースを操作する ​
AWS のアカウントにログインすると，まず最初に表示されるのがAWS コンソールである (figure_title)．
AWSマネージメントコンソール画面
コンソールを使うことで， EC2 のインスタンスを立ち上げたり，S3 のデータを追加・削除したり，ログを閲覧したりなど，AWS 上のあらゆるリソースの操作を GUI (Graphical User Interface) を通して実行することができる． 初めて触る機能をポチポチと試したり，デバッグを行うときなどにとても便利である．
コンソールはさらっと機能を試したり，開発中のクラウドのデバッグをするときには便利なのであるが，実際にクラウドの開発をする場面でこれを直接いじることはあまりない． むしろ，次に紹介する API を使用して，プログラムとしてクラウドのリソースを記述することで開発を行うのが一般的である． そのような理由で，本書では AWS コンソールを使った AWS の使い方はあまり触れない． AWS のドキュメンテーションには，たくさんの チュートリアル が用意されており，コンソール画面から様々な操作を行う方法が記述されているので，興味がある読者はそちらを参照されたい．
API からリソースを操作する ​
API (Application Programming Interface) を使うことで，コマンドを AWS に送信し，クラウドのリソースの操作をすることができる． API とは，端的に言えば AWS が公開しているコマンドの一覧であり，GET, POST, DELETE などの REST API から構成されている (REST API については ??? で簡単に解説する)． が，直接 REST API を入力するのは面倒であるので，その手間を解消するための様々なツールが提供されている．
例えば， AWS CLI は， UNIX コンソールから AWS API を実行するための CLI (Command Line Interface) である． CLI に加えて，いろいろなプログラミング言語での SDK (Software Development Kit) が提供されている．以下に一例を挙げる．
具体的な API の使用例を見てみよう．
S3 に新しい保存領域 (Bucket (バケット) とよばれる) を追加したいとしよう． AWS CLI を使った場合は，次のようなコマンドを打てばよい．
shell
$ aws s3 mb s3://my-bucket --region ap-northeast-1
$ aws s3 mb s3://my-bucket --region ap-northeast-1
上記のコマンドは， my-bucket という名前のバケットを， ap-northeast-1 のリージョンに作成する．
Python からこれと同じ操作を実行するには， boto3 ライブラリを使って，次のようなスクリプトを実行する．
python
import boto3
 
 s3_client = boto3.client("s3", region_name="ap-northeast-1")
 s3_client.create_bucket(Bucket="my-bucket")
import boto3
diff --git a/assets/development_aws_aws-get-started.md.df995644.lean.js b/assets/development_aws_aws-get-started.md.305acfd1.lean.js
similarity index 89%
rename from assets/development_aws_aws-get-started.md.df995644.lean.js
rename to assets/development_aws_aws-get-started.md.305acfd1.lean.js
index 3086858d..9b12e4d6 100644
--- a/assets/development_aws_aws-get-started.md.df995644.lean.js
+++ b/assets/development_aws_aws-get-started.md.305acfd1.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as o,e as p,f as e,g as t,h as r,i as c,j as y}from"./chunks/iac.2263bc12.js";import{_ as F,a as D}from"./chunks/VPC.e1acca4d.js";import{_ as A,c as i,o as C,R as B}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"AWS 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-get-started.md","filePath":"development/aws/aws-get-started.md","lastUpdated":1684858945000}'),u={name:"development/aws/aws-get-started.md"},d=B("",106),m=[d];function b(q,h,g,f,E,S){return C(),i("div",null,m)}const W=A(u,[["render",b]]);export{w as __pageData,W as default};
+import{_ as s,a,b as n,c as l,d as o,e as p,f as e,g as t,h as r,i as c,j as y}from"./chunks/iac.2263bc12.js";import{_ as F,a as D}from"./chunks/VPC.e1acca4d.js";import{_ as A,c as i,o as C,R as B}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"AWS 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/aws-get-started.md","filePath":"development/aws/aws-get-started.md","lastUpdated":1684859551000}'),u={name:"development/aws/aws-get-started.md"},d=B("",106),m=[d];function b(q,h,g,f,E,S){return C(),i("div",null,m)}const W=A(u,[["render",b]]);export{w as __pageData,W as default};
diff --git a/assets/development_aws_closing.md.b2d7e915.js b/assets/development_aws_closing.md.73173906.js
similarity index 89%
rename from assets/development_aws_closing.md.b2d7e915.js
rename to assets/development_aws_closing.md.73173906.js
index 86cf03ec..8077a001 100644
--- a/assets/development_aws_closing.md.b2d7e915.js
+++ b/assets/development_aws_closing.md.73173906.js
@@ -1 +1 @@
-import{_ as a,c as t,o as s,z as e,a as o}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"まとめ","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/closing.md","filePath":"development/aws/closing.md","lastUpdated":1684858945000}'),n={name:"development/aws/closing.md"},c=e("h1",{id:"まとめ",tabindex:"-1"},[o("まとめ "),e("a",{class:"header-anchor",href:"#まとめ","aria-label":'Permalink to "まとめ"'},"​")],-1),r=[c];function d(l,i,p,_,m,h){return s(),t("div",null,r)}const u=a(n,[["render",d]]);export{g as __pageData,u as default};
+import{_ as a,c as t,o as s,z as e,a as o}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"まとめ","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/closing.md","filePath":"development/aws/closing.md","lastUpdated":1684859551000}'),n={name:"development/aws/closing.md"},c=e("h1",{id:"まとめ",tabindex:"-1"},[o("まとめ "),e("a",{class:"header-anchor",href:"#まとめ","aria-label":'Permalink to "まとめ"'},"​")],-1),r=[c];function d(l,i,p,_,m,h){return s(),t("div",null,r)}const u=a(n,[["render",d]]);export{g as __pageData,u as default};
diff --git a/assets/development_aws_closing.md.b2d7e915.lean.js b/assets/development_aws_closing.md.73173906.lean.js
similarity index 89%
rename from assets/development_aws_closing.md.b2d7e915.lean.js
rename to assets/development_aws_closing.md.73173906.lean.js
index 86cf03ec..8077a001 100644
--- a/assets/development_aws_closing.md.b2d7e915.lean.js
+++ b/assets/development_aws_closing.md.73173906.lean.js
@@ -1 +1 @@
-import{_ as a,c as t,o as s,z as e,a as o}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"まとめ","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/closing.md","filePath":"development/aws/closing.md","lastUpdated":1684858945000}'),n={name:"development/aws/closing.md"},c=e("h1",{id:"まとめ",tabindex:"-1"},[o("まとめ "),e("a",{class:"header-anchor",href:"#まとめ","aria-label":'Permalink to "まとめ"'},"​")],-1),r=[c];function d(l,i,p,_,m,h){return s(),t("div",null,r)}const u=a(n,[["render",d]]);export{g as __pageData,u as default};
+import{_ as a,c as t,o as s,z as e,a as o}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"まとめ","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/closing.md","filePath":"development/aws/closing.md","lastUpdated":1684859551000}'),n={name:"development/aws/closing.md"},c=e("h1",{id:"まとめ",tabindex:"-1"},[o("まとめ "),e("a",{class:"header-anchor",href:"#まとめ","aria-label":'Permalink to "まとめ"'},"​")],-1),r=[c];function d(l,i,p,_,m,h){return s(),t("div",null,r)}const u=a(n,[["render",d]]);export{g as __pageData,u as default};
diff --git a/assets/development_aws_cloud.md.34efce50.js b/assets/development_aws_cloud.md.d6418ba0.js
similarity index 99%
rename from assets/development_aws_cloud.md.34efce50.js
rename to assets/development_aws_cloud.md.d6418ba0.js
index 3ea6aa07..5de656de 100644
--- a/assets/development_aws_cloud.md.34efce50.js
+++ b/assets/development_aws_cloud.md.d6418ba0.js
@@ -1 +1 @@
-import{_ as o,a as e,b as r,c as t}from"./chunks/terminal.94539704.js";import{_ as a,c as i,o as p,R as n}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"クラウド概論","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/cloud.md","filePath":"development/aws/cloud.md","lastUpdated":1684858945000}'),s={name:"development/aws/cloud.md"},l=n('
クラウド概論 ​
クラウドとは？ ​
Cloud
クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．
学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．
The NIST Definition of Cloud Computing
これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．
  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．
  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．
  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．
  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．
  • Measured service 計算資源の利用量を計測・監視することができる．
…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．
個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．
クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．
このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．
著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．
クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．
先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．
  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．
  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．
  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．
本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．
SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (???) などの演習を提供する．
なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については ??? などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．
最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．
一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．
第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．
本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．
なぜクラウドを使うのか？ ​
上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．
  1. 自由にサーバーのサイズをスケールできる
    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない
    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0
    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．
    クラウドと自前計算機環境の経済的コストの比較
とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．
ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．
  1. クラウドは賢く使わないといけない
    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ
    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ
    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．
Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？
Terminal
この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．
時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．
最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．
',34),c=[l];function g(u,d,_,f,m,h){return p(),i("div",null,c)}const b=a(s,[["render",g]]);export{T as __pageData,b as default};
+import{_ as o,a as e,b as r,c as t}from"./chunks/terminal.94539704.js";import{_ as a,c as i,o as p,R as n}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"クラウド概論","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/cloud.md","filePath":"development/aws/cloud.md","lastUpdated":1684859551000}'),s={name:"development/aws/cloud.md"},l=n('
クラウド概論 ​
クラウドとは？ ​
Cloud
クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．
学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．
The NIST Definition of Cloud Computing
これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．
  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．
  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．
  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．
  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．
  • Measured service 計算資源の利用量を計測・監視することができる．
…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．
個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．
クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．
このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．
著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．
クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．
先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．
  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．
  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．
  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．
本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．
SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (???) などの演習を提供する．
なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については ??? などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．
最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．
一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．
第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．
本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．
なぜクラウドを使うのか？ ​
上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．
  1. 自由にサーバーのサイズをスケールできる
    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない
    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0
    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．
    クラウドと自前計算機環境の経済的コストの比較
とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．
ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．
  1. クラウドは賢く使わないといけない
    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ
    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ
    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．
Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？
Terminal
この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．
時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．
最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．
',34),c=[l];function g(u,d,_,f,m,h){return p(),i("div",null,c)}const b=a(s,[["render",g]]);export{T as __pageData,b as default};
diff --git a/assets/development_aws_cloud.md.34efce50.lean.js b/assets/development_aws_cloud.md.d6418ba0.lean.js
similarity index 87%
rename from assets/development_aws_cloud.md.34efce50.lean.js
rename to assets/development_aws_cloud.md.d6418ba0.lean.js
index 0339637e..a7f1b0aa 100644
--- a/assets/development_aws_cloud.md.34efce50.lean.js
+++ b/assets/development_aws_cloud.md.d6418ba0.lean.js
@@ -1 +1 @@
-import{_ as o,a as e,b as r,c as t}from"./chunks/terminal.94539704.js";import{_ as a,c as i,o as p,R as n}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"クラウド概論","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/cloud.md","filePath":"development/aws/cloud.md","lastUpdated":1684858945000}'),s={name:"development/aws/cloud.md"},l=n("",34),c=[l];function g(u,d,_,f,m,h){return p(),i("div",null,c)}const b=a(s,[["render",g]]);export{T as __pageData,b as default};
+import{_ as o,a as e,b as r,c as t}from"./chunks/terminal.94539704.js";import{_ as a,c as i,o as p,R as n}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"クラウド概論","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/cloud.md","filePath":"development/aws/cloud.md","lastUpdated":1684859551000}'),s={name:"development/aws/cloud.md"},l=n("",34),c=[l];function g(u,d,_,f,m,h){return p(),i("div",null,c)}const b=a(s,[["render",g]]);export{T as __pageData,b as default};
diff --git a/assets/development_aws_docker-system.md.209ef2e6.js b/assets/development_aws_docker-system.md.b4b42fe0.js
similarity index 99%
rename from assets/development_aws_docker-system.md.209ef2e6.js
rename to assets/development_aws_docker-system.md.b4b42fe0.js
index 6f6208ca..26ee1a37 100644
--- a/assets/development_aws_docker-system.md.209ef2e6.js
+++ b/assets/development_aws_docker-system.md.b4b42fe0.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as e,d as o,e as p,f as l,g as r}from"./chunks/ecs.73d77e6a.js";import{_ as t,c,o as i,R as y}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Docker 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/docker-system.md","filePath":"development/aws/docker-system.md","lastUpdated":1684858945000}'),u={name:"development/aws/docker-system.md"},C=y('
Docker 入門 ​
ここまでの章で扱ってきたハンズオンでは，単一のサーバーを立ち上げ，それに SSH でログインをして，コマンドを叩くことで計算を行ってきた． いわば，パーソナルコンピュータの延長のような形でクラウドを使ってきたわけである． このような，インターネットのどこからでもアクセスできるパーソナルコンピュータとしてのクラウドという使い方も，もちろん便利であるし，いろいろな応用の可能性がある． しかし，これだけではクラウドの本当の価値は十分に発揮されていないと言うべきだろう． ??? で述べたように，現代的なクラウドの一番の強みは自由に計算機の規模を拡大できることにある． すなわち，多数のサーバーを同時に起動し，複数のジョブを分散並列的に実行させることで大量のデータを処理してこそ，クラウドの本領が発揮されるのである．
本章からはじまる 3 章分 (Docker 入門, ???, ???) を使って，クラウドを利用することでどのように大規模な計算システムを構築しビッグデータの解析に立ち向かうのか，その片鱗をお見せしたい． とくに，前章で扱った深層学習をどのようにビッグデータに適用していくかという点に焦点を絞って議論していきたい． そのための前準備として，本章では Docker とよばれる計算機環境の仮想化ソフトウェアを紹介する (figure_title)． 現代のクラウドは Docker なしには成り立たないといっても過言ではないだろう． クラウドに限らず，ローカルで行う計算処理にも Docker は大変便利である． AWS からは少し話が離れるが，しっかりと理解して前に進んでもらいたい．
機械学習の大規模化 ​
先ほどから"計算システムの大規模化"と繰り返し唱えているが，それは具体的にはどのようなものを指しているのか？ ここでは大規模データを処理するための計算機システムを，機械学習を例にとって見てみよう．
??? で紹介した GPT-3 のような，超巨大な数のパラメータを有する深層学習モデルを学習させたいとしよう． そのような計算を行いたい場合，一つのサーバーでは計算力が到底足りない． したがって，典型的には figure_title に示すような計算システムの設計がなされる． すなわち，大量の教師データを小さなチャンクとして複数のマシンに分散し，並列的にニューラルネットのパラメータを最適化していくという構造である．
複数の計算機を使った大規模な深層学習モデルの訓練
あるいは，学習済みのモデルを大量のデータに適用し，解析を行いたいとしよう． たとえば， SNS のプラットフォームで大量の画像が与えられて，それぞれの写真に何が写っているのかをラベルづけする，などのアプリケーションを想定できる． そのような場合は， figure_title のようなアーキテクチャが考えられるだろう． すなわち，大量のデータを複数のマシンで分割し，それぞれのマシンで推論の計算を行うというような構造である．
複数の計算機による深層学習モデルを使った推論計算
このような複数の計算機を同時に走らせるようなアプリケーションをクラウド上で実現するには，どのようにすればよいのだろうか？
重要なポイントとして， figure_title や figure_title で起動している複数のマシンは，基本的に全く同一の OS・計算環境を有している点である． ここで，個人のコンピュータで行うようなインストールの操作を，各マシンで行うこともできるが，それは大変な手間であるし，メンテナンスも面倒だろう． すなわち，大規模な計算システムを構築するには，簡単に計算環境を複製できるような仕組みが必要であるということがわかる．
そのような目的を実現するために使われるのが， Docker とよばれるソフトウェアである．
Docker とは ​
Docker のアイコン
Docker とは， コンテナ (Container) とよばれる仮想環境下で，ホスト OS とは独立した別の計算環境を走らせるためのソフトウェアである． Docker を使うことで， OS を含めたすべてのプログラムをコンパクトにパッケージングすることが可能になる (パッケージされた一つの計算環境のことを **イメージ (Image)**とよぶ)． Docker を使うことで，クラウドのサーバー上に瞬時に計算環境を複製することが可能になり， figure_title で見たような複数の計算機を同時に走らせるためのシステムが実現できる．
Docker は 2013 年に Solomon Hykes らを中心に開発され，それ以降爆発的に普及し，クラウドコンピューティングだけでなく，機械学習・科学計算の文脈などでも欠かすことのできないソフトウェアとなった． Docker はエンタープライズ向けの製品を除き無料で使用することができ，コアの部分は オープンソースプロジェクト として公開されている． Docker は Linux, Windows, Mac いずれの OS でも提供されている． 概念としては， Docker は仮想マシン (Virtual machine; VM) にとても近い． ここでは， VM との対比をしながら，Docker とはなにかを簡単に説明しよう．
仮想マシン (VM) とは，ホストとなるマシンの上に，仮想化された OS を走らせる技術である (figure_title)． VM には ハイパーバイザー (Hypervisor) とよばれるレイヤーが存在する． Hypervisor はまず，物理的な計算機リソース (CPU, RAM, network など) を分割し，仮想化する． たとえば， ホストマシンに物理的な CPU が 4 コアあるとして，ハイパーバイザーはそれを (2,2) 個の組に仮想的に分割することができる． VM 上で起動する OS には，ハイパーバイザーによって仮想化されたハードウェアが割り当てられる． VM 上で起動する OS は基本的に完全に独立であり，たとえば OS-A は OS-B に割り当てられた CPU やメモリー領域にアクセスすることはできない (これを isolation とよぶ)． VM を作成するための有名なソフトウェアとしては， VMware， VirtualBox， Xen などがある． また，これまで触ってきた EC2 も，基本的に VM 技術を使うことで所望のスペックをもった仮想マシンがユーザーに提示される．
Docker も， VM と同様に，仮想化された OS をホストの OS 上に走らせるための技術である． VM に対し， Docker ではハードウェアレベルの仮想化は行われておらず，すべての仮想化はソフトウェアレベルで実現されている (figure_title)． Docker で走る仮想 OS は，多くの部分をホストの OS に依存しており，結果として非常にコンパクトである． その結果， Docker で仮想 OS を起動するために要する時間は， VM に比べて圧倒的に早い． また， パッケージ化された環境 (=イメージ) のサイズも完全な OS に比べ圧倒的に小さくなるので，ネットワークを通じたやり取りが非常に高速化される点も重要である 加えて， VM のいくつかの実装では，メタル (仮想化マシンに対して，物理的なハードウェア上で直接起動する場合のこと) と比べ，ハイパーバイザーレイヤーでのオーバーヘッドなどにより性能が低下することが知られているが， Docker ではメタルとほぼ同様の性能を引き出すことができるとされている．
その他， VM との相違点などはたくさんあるのだが，ここではこれ以上詳細には立ち入らない． 大事なのは， Docker とはとてもコンパクトかつハイパフォーマンスな仮想計算環境を作るツールである，という点である． その手軽さゆえに，2013 年の登場以降，クラウドシステムでの利用が急速に増加し，現代のクラウドでは欠くことのできない中心的な技術になっている．
Docker (左) と VM (右) の比較 (画像出典: https://www.docker.com/blog/containers-replacing-virtual-machines/)
職業的プログラマーにとっての"三種の神器"とはなんだろうか？ 多様な意見があると思うが，筆者は Git, Vim そして Docker を挙げたい．
Git は多くの読者がご存じの通り，コードの変更を追跡するためのシステムである． Linux の作成者である Linus Torvalds によって 2005 年に誕生した． チームでの開発を進める際には欠かせないツールだ．
Vim は 1991 年から 30 年以上の間プログラマーたちに愛されてきたテキストエディターである． Stackoverflow が行った 2019 年のアンケート によると，開発環境の部門で 5 位の人気を獲得している． たくさんのショートカットと様々なカスタム設定が提供されているので，初見の人にはなかなかハードルが高いが，一度マスターすれば他のモダンなエディターや統合開発環境に負けない，あるいはそれ以上の開発体験を実現することができる．
これらの十年以上の歴史あるツールに並んで，第三番目の三種の神器として挙げたいのが Docker だ． Docker はプログラマーの開発のワークフローを一変させた． たとえば，プロジェクトごとに Docker イメージを作成することで，どの OS・コンピュータ でも全く同じ計算環境で開発・テストを実行することができるようになった． また， DevOps や CI / CD (Continuous Integration / Continuous Deployment) といった最近の開発ワークフローも Docker のようなコンテナ技術の存在に立脚している． さらにはサーバーレスコンピューティング (???) といった概念も，コンテナ技術の生んだ大きな技術革新といえる．
あなたにとっての三種の神器はなんだろうか？ また，これからの未来ではどんな新しいツールが三種の神器としてプログラマーのワークフローを革新していくだろうか？
Docker チュートリアル ​
Docker とはなにかを理解するためには，実際に触って動かしてみるのが一番有効な手立てである． ここでは， Docker の簡単なチュートリアルを行っていく．
Docker のインストールについては， ??? および 公式のドキュメンテーション を参照してもらいたい． Docker のインストールが完了している前提で，以下は話を進めるものとする．
Docker 用語集 ​
Docker を使い始めるに当たり，最初に主要な用語を解説しよう． 次のパラグラフで太字で強調された用語を頭に入れた上で，続くチュートリアルに取り組んでいただきたい．
Docker を起動する際の大まかなステップを示したのが figure_title である． パッケージされた一つの計算環境のことを **イメージ (Image)**とよぶ． イメージは， Docker Hub などのリポジトリで配布されているものをダウンロードするか，自分でカスタムのイメージを作成することも可能である． イメージを作成するための”レシピ”を記述したファイルが Dockerfile である． Dockerfile からイメージを作成する操作を build とよぶ． イメージがホストマシンのメモリにロードされ，起動状態にある計算環境のことを コンテナ (Container) とよぶ． Container を起動するために使用されるコマンドが run である．
Image と Container
イメージをダウンロード ​
パッケージ化された Docker の仮想環境 (= イメージ (Image)) は， Docker Hub からダウンロードできる． Docker Hub には，個人や企業・団体が作成した Docker イメージが集められており， GitHub などと同じ感覚で，オープンな形で公開されている．
たとえば， Ubuntu のイメージは Ubuntu の公式リポジトリ で公開されており， pull コマンドを使うことでローカルにダウンロードすることができる．
shell
$ docker pull ubuntu:18.04
$ docker pull ubuntu:18.04
ここで，イメージ名の : (コロン) 以降に続く文字列を タグ (tag) と呼び，主にバージョンを指定するなどの目的で使われる．
pull コマンドはデフォルトでは Docker Hub でイメージを検索し，ダウンロードを行う． Docker イメージを公開するためのデータベース (レジストリ (registry) とよぶ) は Docker Hub だけではなく，たとえば GitLab や GitHub は独自のレジストリ機能を提供しているし，個人のサーバーでレジストリを立ち上げることも可能である． Docker Hub 以外のレジストリから pull するには， myregistry.local:5000/testing/test-image のように，イメージ名の先頭につける形でレジストリのアドレス (さらにオプションとしてポート番号) を指定する．
コンテナを起動 ​
Pull してきたイメージを起動するには， run コマンドを使う．
shell
$ docker run -it ubuntu:18.04
$ docker run -it ubuntu:18.04
ここで， -it とは，インタラクティブな shell のセッションを開始するために必要なオプションである．
このコマンドを実行すると，仮想化された Ubuntu が起動され，コマンドラインからコマンドが打ち込めるようになる (figure_title)． このように起動状態にある計算環境 (ランタイム) のことを Container (コンテナ) とよぶ．
Docker を使って ubuntu:18.04 イメージを起動
ここで使用した ubuntu:18.04 のイメージは，空の Ubuntu OS だが，すでにプログラムがインストール済みのものもある． これは， ??? でみた DLAMI と概念として似ている． たとえば， PyTorch がインストール済みのイメージは PyTorch 公式の Docker Hub リポジトリ で公開されている．
これを起動してみよう．
shell
$ docker run -it pytorch/pytorch
$ docker run -it pytorch/pytorch
docker run を実行したとき，ローカルに該当するイメージが見つからない場合は，自動的に Docker Hub からダウンロードされる．
pytorch のコンテナが起動したら， Python のシェルを立ち上げて， pytorch をインポートしてみよう．
shell
$ python3
+import{_ as s,a,b as n,c as e,d as o,e as p,f as l,g as r}from"./chunks/ecs.73d77e6a.js";import{_ as t,c,o as i,R as y}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Docker 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/docker-system.md","filePath":"development/aws/docker-system.md","lastUpdated":1684859551000}'),u={name:"development/aws/docker-system.md"},C=y('
Docker 入門 ​
ここまでの章で扱ってきたハンズオンでは，単一のサーバーを立ち上げ，それに SSH でログインをして，コマンドを叩くことで計算を行ってきた． いわば，パーソナルコンピュータの延長のような形でクラウドを使ってきたわけである． このような，インターネットのどこからでもアクセスできるパーソナルコンピュータとしてのクラウドという使い方も，もちろん便利であるし，いろいろな応用の可能性がある． しかし，これだけではクラウドの本当の価値は十分に発揮されていないと言うべきだろう． ??? で述べたように，現代的なクラウドの一番の強みは自由に計算機の規模を拡大できることにある． すなわち，多数のサーバーを同時に起動し，複数のジョブを分散並列的に実行させることで大量のデータを処理してこそ，クラウドの本領が発揮されるのである．
本章からはじまる 3 章分 (Docker 入門, ???, ???) を使って，クラウドを利用することでどのように大規模な計算システムを構築しビッグデータの解析に立ち向かうのか，その片鱗をお見せしたい． とくに，前章で扱った深層学習をどのようにビッグデータに適用していくかという点に焦点を絞って議論していきたい． そのための前準備として，本章では Docker とよばれる計算機環境の仮想化ソフトウェアを紹介する (figure_title)． 現代のクラウドは Docker なしには成り立たないといっても過言ではないだろう． クラウドに限らず，ローカルで行う計算処理にも Docker は大変便利である． AWS からは少し話が離れるが，しっかりと理解して前に進んでもらいたい．
機械学習の大規模化 ​
先ほどから"計算システムの大規模化"と繰り返し唱えているが，それは具体的にはどのようなものを指しているのか？ ここでは大規模データを処理するための計算機システムを，機械学習を例にとって見てみよう．
??? で紹介した GPT-3 のような，超巨大な数のパラメータを有する深層学習モデルを学習させたいとしよう． そのような計算を行いたい場合，一つのサーバーでは計算力が到底足りない． したがって，典型的には figure_title に示すような計算システムの設計がなされる． すなわち，大量の教師データを小さなチャンクとして複数のマシンに分散し，並列的にニューラルネットのパラメータを最適化していくという構造である．
複数の計算機を使った大規模な深層学習モデルの訓練
あるいは，学習済みのモデルを大量のデータに適用し，解析を行いたいとしよう． たとえば， SNS のプラットフォームで大量の画像が与えられて，それぞれの写真に何が写っているのかをラベルづけする，などのアプリケーションを想定できる． そのような場合は， figure_title のようなアーキテクチャが考えられるだろう． すなわち，大量のデータを複数のマシンで分割し，それぞれのマシンで推論の計算を行うというような構造である．
複数の計算機による深層学習モデルを使った推論計算
このような複数の計算機を同時に走らせるようなアプリケーションをクラウド上で実現するには，どのようにすればよいのだろうか？
重要なポイントとして， figure_title や figure_title で起動している複数のマシンは，基本的に全く同一の OS・計算環境を有している点である． ここで，個人のコンピュータで行うようなインストールの操作を，各マシンで行うこともできるが，それは大変な手間であるし，メンテナンスも面倒だろう． すなわち，大規模な計算システムを構築するには，簡単に計算環境を複製できるような仕組みが必要であるということがわかる．
そのような目的を実現するために使われるのが， Docker とよばれるソフトウェアである．
Docker とは ​
Docker のアイコン
Docker とは， コンテナ (Container) とよばれる仮想環境下で，ホスト OS とは独立した別の計算環境を走らせるためのソフトウェアである． Docker を使うことで， OS を含めたすべてのプログラムをコンパクトにパッケージングすることが可能になる (パッケージされた一つの計算環境のことを **イメージ (Image)**とよぶ)． Docker を使うことで，クラウドのサーバー上に瞬時に計算環境を複製することが可能になり， figure_title で見たような複数の計算機を同時に走らせるためのシステムが実現できる．
Docker は 2013 年に Solomon Hykes らを中心に開発され，それ以降爆発的に普及し，クラウドコンピューティングだけでなく，機械学習・科学計算の文脈などでも欠かすことのできないソフトウェアとなった． Docker はエンタープライズ向けの製品を除き無料で使用することができ，コアの部分は オープンソースプロジェクト として公開されている． Docker は Linux, Windows, Mac いずれの OS でも提供されている． 概念としては， Docker は仮想マシン (Virtual machine; VM) にとても近い． ここでは， VM との対比をしながら，Docker とはなにかを簡単に説明しよう．
仮想マシン (VM) とは，ホストとなるマシンの上に，仮想化された OS を走らせる技術である (figure_title)． VM には ハイパーバイザー (Hypervisor) とよばれるレイヤーが存在する． Hypervisor はまず，物理的な計算機リソース (CPU, RAM, network など) を分割し，仮想化する． たとえば， ホストマシンに物理的な CPU が 4 コアあるとして，ハイパーバイザーはそれを (2,2) 個の組に仮想的に分割することができる． VM 上で起動する OS には，ハイパーバイザーによって仮想化されたハードウェアが割り当てられる． VM 上で起動する OS は基本的に完全に独立であり，たとえば OS-A は OS-B に割り当てられた CPU やメモリー領域にアクセスすることはできない (これを isolation とよぶ)． VM を作成するための有名なソフトウェアとしては， VMware， VirtualBox， Xen などがある． また，これまで触ってきた EC2 も，基本的に VM 技術を使うことで所望のスペックをもった仮想マシンがユーザーに提示される．
Docker も， VM と同様に，仮想化された OS をホストの OS 上に走らせるための技術である． VM に対し， Docker ではハードウェアレベルの仮想化は行われておらず，すべての仮想化はソフトウェアレベルで実現されている (figure_title)． Docker で走る仮想 OS は，多くの部分をホストの OS に依存しており，結果として非常にコンパクトである． その結果， Docker で仮想 OS を起動するために要する時間は， VM に比べて圧倒的に早い． また， パッケージ化された環境 (=イメージ) のサイズも完全な OS に比べ圧倒的に小さくなるので，ネットワークを通じたやり取りが非常に高速化される点も重要である 加えて， VM のいくつかの実装では，メタル (仮想化マシンに対して，物理的なハードウェア上で直接起動する場合のこと) と比べ，ハイパーバイザーレイヤーでのオーバーヘッドなどにより性能が低下することが知られているが， Docker ではメタルとほぼ同様の性能を引き出すことができるとされている．
その他， VM との相違点などはたくさんあるのだが，ここではこれ以上詳細には立ち入らない． 大事なのは， Docker とはとてもコンパクトかつハイパフォーマンスな仮想計算環境を作るツールである，という点である． その手軽さゆえに，2013 年の登場以降，クラウドシステムでの利用が急速に増加し，現代のクラウドでは欠くことのできない中心的な技術になっている．
Docker (左) と VM (右) の比較 (画像出典: https://www.docker.com/blog/containers-replacing-virtual-machines/)
職業的プログラマーにとっての"三種の神器"とはなんだろうか？ 多様な意見があると思うが，筆者は Git, Vim そして Docker を挙げたい．
Git は多くの読者がご存じの通り，コードの変更を追跡するためのシステムである． Linux の作成者である Linus Torvalds によって 2005 年に誕生した． チームでの開発を進める際には欠かせないツールだ．
Vim は 1991 年から 30 年以上の間プログラマーたちに愛されてきたテキストエディターである． Stackoverflow が行った 2019 年のアンケート によると，開発環境の部門で 5 位の人気を獲得している． たくさんのショートカットと様々なカスタム設定が提供されているので，初見の人にはなかなかハードルが高いが，一度マスターすれば他のモダンなエディターや統合開発環境に負けない，あるいはそれ以上の開発体験を実現することができる．
これらの十年以上の歴史あるツールに並んで，第三番目の三種の神器として挙げたいのが Docker だ． Docker はプログラマーの開発のワークフローを一変させた． たとえば，プロジェクトごとに Docker イメージを作成することで，どの OS・コンピュータ でも全く同じ計算環境で開発・テストを実行することができるようになった． また， DevOps や CI / CD (Continuous Integration / Continuous Deployment) といった最近の開発ワークフローも Docker のようなコンテナ技術の存在に立脚している． さらにはサーバーレスコンピューティング (???) といった概念も，コンテナ技術の生んだ大きな技術革新といえる．
あなたにとっての三種の神器はなんだろうか？ また，これからの未来ではどんな新しいツールが三種の神器としてプログラマーのワークフローを革新していくだろうか？
Docker チュートリアル ​
Docker とはなにかを理解するためには，実際に触って動かしてみるのが一番有効な手立てである． ここでは， Docker の簡単なチュートリアルを行っていく．
Docker のインストールについては， ??? および 公式のドキュメンテーション を参照してもらいたい． Docker のインストールが完了している前提で，以下は話を進めるものとする．
Docker 用語集 ​
Docker を使い始めるに当たり，最初に主要な用語を解説しよう． 次のパラグラフで太字で強調された用語を頭に入れた上で，続くチュートリアルに取り組んでいただきたい．
Docker を起動する際の大まかなステップを示したのが figure_title である． パッケージされた一つの計算環境のことを **イメージ (Image)**とよぶ． イメージは， Docker Hub などのリポジトリで配布されているものをダウンロードするか，自分でカスタムのイメージを作成することも可能である． イメージを作成するための”レシピ”を記述したファイルが Dockerfile である． Dockerfile からイメージを作成する操作を build とよぶ． イメージがホストマシンのメモリにロードされ，起動状態にある計算環境のことを コンテナ (Container) とよぶ． Container を起動するために使用されるコマンドが run である．
Image と Container
イメージをダウンロード ​
パッケージ化された Docker の仮想環境 (= イメージ (Image)) は， Docker Hub からダウンロードできる． Docker Hub には，個人や企業・団体が作成した Docker イメージが集められており， GitHub などと同じ感覚で，オープンな形で公開されている．
たとえば， Ubuntu のイメージは Ubuntu の公式リポジトリ で公開されており， pull コマンドを使うことでローカルにダウンロードすることができる．
shell
$ docker pull ubuntu:18.04
$ docker pull ubuntu:18.04
ここで，イメージ名の : (コロン) 以降に続く文字列を タグ (tag) と呼び，主にバージョンを指定するなどの目的で使われる．
pull コマンドはデフォルトでは Docker Hub でイメージを検索し，ダウンロードを行う． Docker イメージを公開するためのデータベース (レジストリ (registry) とよぶ) は Docker Hub だけではなく，たとえば GitLab や GitHub は独自のレジストリ機能を提供しているし，個人のサーバーでレジストリを立ち上げることも可能である． Docker Hub 以外のレジストリから pull するには， myregistry.local:5000/testing/test-image のように，イメージ名の先頭につける形でレジストリのアドレス (さらにオプションとしてポート番号) を指定する．
コンテナを起動 ​
Pull してきたイメージを起動するには， run コマンドを使う．
shell
$ docker run -it ubuntu:18.04
$ docker run -it ubuntu:18.04
ここで， -it とは，インタラクティブな shell のセッションを開始するために必要なオプションである．
このコマンドを実行すると，仮想化された Ubuntu が起動され，コマンドラインからコマンドが打ち込めるようになる (figure_title)． このように起動状態にある計算環境 (ランタイム) のことを Container (コンテナ) とよぶ．
Docker を使って ubuntu:18.04 イメージを起動
ここで使用した ubuntu:18.04 のイメージは，空の Ubuntu OS だが，すでにプログラムがインストール済みのものもある． これは， ??? でみた DLAMI と概念として似ている． たとえば， PyTorch がインストール済みのイメージは PyTorch 公式の Docker Hub リポジトリ で公開されている．
これを起動してみよう．
shell
$ docker run -it pytorch/pytorch
$ docker run -it pytorch/pytorch
docker run を実行したとき，ローカルに該当するイメージが見つからない場合は，自動的に Docker Hub からダウンロードされる．
pytorch のコンテナが起動したら， Python のシェルを立ち上げて， pytorch をインポートしてみよう．
shell
$ python3
 Python 3.7.7 (default, May  7 2020, 21:25:33)
 [GCC 7.3.0] :: Anaconda, Inc. on linux
 Type "help", "copyright", "credits" or "license" for more information.
diff --git a/assets/development_aws_docker-system.md.209ef2e6.lean.js b/assets/development_aws_docker-system.md.b4b42fe0.lean.js
similarity index 88%
rename from assets/development_aws_docker-system.md.209ef2e6.lean.js
rename to assets/development_aws_docker-system.md.b4b42fe0.lean.js
index 6d21cbf8..e9e5a49b 100644
--- a/assets/development_aws_docker-system.md.209ef2e6.lean.js
+++ b/assets/development_aws_docker-system.md.b4b42fe0.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as e,d as o,e as p,f as l,g as r}from"./chunks/ecs.73d77e6a.js";import{_ as t,c,o as i,R as y}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Docker 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/docker-system.md","filePath":"development/aws/docker-system.md","lastUpdated":1684858945000}'),u={name:"development/aws/docker-system.md"},C=y("",73),d=[C];function h(m,g,A,D,k,b){return i(),c("div",null,d)}const _=t(u,[["render",h]]);export{f as __pageData,_ as default};
+import{_ as s,a,b as n,c as e,d as o,e as p,f as l,g as r}from"./chunks/ecs.73d77e6a.js";import{_ as t,c,o as i,R as y}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Docker 入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/docker-system.md","filePath":"development/aws/docker-system.md","lastUpdated":1684859551000}'),u={name:"development/aws/docker-system.md"},C=y("",73),d=[C];function h(m,g,A,D,k,b){return i(),c("div",null,d)}const _=t(u,[["render",h]]);export{f as __pageData,_ as default};
diff --git a/assets/development_aws_handson-bashoutter.md.4863f098.js b/assets/development_aws_handson-bashoutter.md.91af8955.js
similarity index 99%
rename from assets/development_aws_handson-bashoutter.md.4863f098.js
rename to assets/development_aws_handson-bashoutter.md.91af8955.js
index 6189baa8..55c4dedf 100644
--- a/assets/development_aws_handson-bashoutter.md.4863f098.js
+++ b/assets/development_aws_handson-bashoutter.md.91af8955.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as c}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as r,c as y,o as F,R as i}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Hands-on #6: Bashoutter","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-bashoutter.md","filePath":"development/aws/handson-bashoutter.md","lastUpdated":1684858945000}'),A={name:"development/aws/handson-bashoutter.md"},B=i('
Hands-on #6: Bashoutter ​
さて，最後のハンズオンとなる第六回では，これまで学んできたサーバーレスクラウドの技術を使って，簡単なウェブサービスを作ってみよう． 具体的には，人々が自分の作った俳句を投稿する SNS サービス (Bashoutter と名付ける) を作成してみよう． Lambda, DynamoDB, S3 などの技術をすべて盛り込み，シンプルながらもサーバーレスの利点を生かしたスケーラブルな SNS アプリが誕生する． 最終的には， figure_title のような，ミニマルではあるがとても現代風な SNS サイトが完成する！
ハンズオン#6で作製する SNS アプリケーション "Bashoutter"
準備 ​
ハンズオンのソースコードは GitHub の handson/bashoutter に置いてある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． それ以外に必要な準備はない．
このハンズオンは，基本的に AWS の無料枠 の範囲内で実行できる．
アプリケーションの説明 ​
API ​
今回のアプリケーションでは，人々からの俳句の投稿を受け付けたり，投稿された俳句の一覧を取得する，といった機能を実装したい． この機能を実現するための最小限の設計として， table_title に示すような四つの REST API を今回は実装する． 俳句を投稿する，閲覧する，削除するという基本的なデータ操作を行うための API が完備されている． また， PATCH /haiku/{item_id} は， {item_id} で指定された俳句に”いいね”をするために使用する．
Bashoutter API
GET /haiku
俳句の一覧を取得する
POST /haiku
新しい俳句を投稿する
PATCH /haiku/{item_id}
{item_id} で指定された俳句にお気に入り票を一つ入れる
DELETE /haiku/{item_id}
{item_id} で指定された俳句を削除する
それぞれの API のパラメータおよび返り値の詳細は，ハンズオンのソースコードの中の swagger.yml に定義してある．
Open API Specification (OAS; 少し前は Swagger Specification とよばれていた) は， REST API のための記述フォーマットである． OAS に従って API の仕様が記述されていると，簡単にドキュメンテーションを生成したり，クライアントアプリケーションを自動生成することができる． 今回用意した API 仕様 も， OAS に従って書いてある． 詳しくは Swagger の公式ドキュメンテーション などを参照．
アプリケーションアーキテクチャ ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#5で作製するアプリケーションのアーキテクチャ
簡単にまとめると，次のような設計である．
  • クライアントからの API リクエストは， API Gateway (後述)にまず送信され， API の URI で指定された Lambda 関数へ転送される．
  • それぞれの API のパス (リソース) ごとに独立した Lambda を用意する．
  • 俳句の情報 (作者，本文，投稿日時など) を記録するためのデータベース (DynamoDB) を用意する．
  • 各 Lambda 関数には， DynamoDB へのアクセス権を付与する．
  • 最後に，ウェブブラウザからコンテンツを表示できるよう， ウェブページの静的コンテンツを配信するための S3 バケットを用意する．クライアントはこの S3 バケットにアクセスすることで HTML/CSS/JS などのコンテンツを取得する．
それでは，プログラムのソースコードを見てみよう (handson/bashoutter/app.py)．
python
class Bashoutter(core.Stack):
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as c}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as r,c as y,o as F,R as i}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Hands-on #6: Bashoutter","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-bashoutter.md","filePath":"development/aws/handson-bashoutter.md","lastUpdated":1684859551000}'),A={name:"development/aws/handson-bashoutter.md"},B=i('
Hands-on #6: Bashoutter ​
さて，最後のハンズオンとなる第六回では，これまで学んできたサーバーレスクラウドの技術を使って，簡単なウェブサービスを作ってみよう． 具体的には，人々が自分の作った俳句を投稿する SNS サービス (Bashoutter と名付ける) を作成してみよう． Lambda, DynamoDB, S3 などの技術をすべて盛り込み，シンプルながらもサーバーレスの利点を生かしたスケーラブルな SNS アプリが誕生する． 最終的には， figure_title のような，ミニマルではあるがとても現代風な SNS サイトが完成する！
ハンズオン#6で作製する SNS アプリケーション "Bashoutter"
準備 ​
ハンズオンのソースコードは GitHub の handson/bashoutter に置いてある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． それ以外に必要な準備はない．
このハンズオンは，基本的に AWS の無料枠 の範囲内で実行できる．
アプリケーションの説明 ​
API ​
今回のアプリケーションでは，人々からの俳句の投稿を受け付けたり，投稿された俳句の一覧を取得する，といった機能を実装したい． この機能を実現するための最小限の設計として， table_title に示すような四つの REST API を今回は実装する． 俳句を投稿する，閲覧する，削除するという基本的なデータ操作を行うための API が完備されている． また， PATCH /haiku/{item_id} は， {item_id} で指定された俳句に”いいね”をするために使用する．
Bashoutter API
GET /haiku
俳句の一覧を取得する
POST /haiku
新しい俳句を投稿する
PATCH /haiku/{item_id}
{item_id} で指定された俳句にお気に入り票を一つ入れる
DELETE /haiku/{item_id}
{item_id} で指定された俳句を削除する
それぞれの API のパラメータおよび返り値の詳細は，ハンズオンのソースコードの中の swagger.yml に定義してある．
Open API Specification (OAS; 少し前は Swagger Specification とよばれていた) は， REST API のための記述フォーマットである． OAS に従って API の仕様が記述されていると，簡単にドキュメンテーションを生成したり，クライアントアプリケーションを自動生成することができる． 今回用意した API 仕様 も， OAS に従って書いてある． 詳しくは Swagger の公式ドキュメンテーション などを参照．
アプリケーションアーキテクチャ ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#5で作製するアプリケーションのアーキテクチャ
簡単にまとめると，次のような設計である．
  • クライアントからの API リクエストは， API Gateway (後述)にまず送信され， API の URI で指定された Lambda 関数へ転送される．
  • それぞれの API のパス (リソース) ごとに独立した Lambda を用意する．
  • 俳句の情報 (作者，本文，投稿日時など) を記録するためのデータベース (DynamoDB) を用意する．
  • 各 Lambda 関数には， DynamoDB へのアクセス権を付与する．
  • 最後に，ウェブブラウザからコンテンツを表示できるよう， ウェブページの静的コンテンツを配信するための S3 バケットを用意する．クライアントはこの S3 バケットにアクセスすることで HTML/CSS/JS などのコンテンツを取得する．
それでは，プログラムのソースコードを見てみよう (handson/bashoutter/app.py)．
python
class Bashoutter(core.Stack):
 
     def __init__(self, scope: core.App, name: str, **kwargs) -> None:
         super().__init__(scope, name, **kwargs)
diff --git a/assets/development_aws_handson-bashoutter.md.4863f098.lean.js b/assets/development_aws_handson-bashoutter.md.91af8955.lean.js
similarity index 89%
rename from assets/development_aws_handson-bashoutter.md.4863f098.lean.js
rename to assets/development_aws_handson-bashoutter.md.91af8955.lean.js
index 4cb31f89..93f531aa 100644
--- a/assets/development_aws_handson-bashoutter.md.4863f098.lean.js
+++ b/assets/development_aws_handson-bashoutter.md.91af8955.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as c}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as r,c as y,o as F,R as i}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Hands-on #6: Bashoutter","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-bashoutter.md","filePath":"development/aws/handson-bashoutter.md","lastUpdated":1684858945000}'),A={name:"development/aws/handson-bashoutter.md"},B=i("",127),D=[B];function d(u,C,b,m,h,_){return F(),y("div",null,D)}const k=r(A,[["render",d]]);export{g as __pageData,k as default};
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as c}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as r,c as y,o as F,R as i}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Hands-on #6: Bashoutter","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-bashoutter.md","filePath":"development/aws/handson-bashoutter.md","lastUpdated":1684859551000}'),A={name:"development/aws/handson-bashoutter.md"},B=i("",127),D=[B];function d(u,C,b,m,h,_){return F(),y("div",null,D)}const k=r(A,[["render",d]]);export{g as __pageData,k as default};
diff --git a/assets/development_aws_handson-ec2.md.64c416d6.js b/assets/development_aws_handson-ec2.md.d527a33c.js
similarity index 99%
rename from assets/development_aws_handson-ec2.md.64c416d6.js
rename to assets/development_aws_handson-ec2.md.d527a33c.js
index 5bc04e5f..07324542 100644
--- a/assets/development_aws_handson-ec2.md.64c416d6.js
+++ b/assets/development_aws_handson-ec2.md.d527a33c.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e}from"./chunks/ec2_keypair_console.fc89ef69.js";import{a as t,_ as c}from"./chunks/VPC.e1acca4d.js";import{_ as r,c as y,o as C,R as A}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Hands-on #1: 初めての EC2 インスタンスを起動する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-ec2.md","filePath":"development/aws/handson-ec2.md","lastUpdated":1684858945000}'),i={name:"development/aws/handson-ec2.md"},F=A('
Hands-on #1: 初めての EC2 インスタンスを起動する ​
ハンズオンの第一回では， CDK を使って EC2 のインスタンス(仮想サーバー)を作成し，SSH でサーバーにログインする，という演習を行う． このハンズオンを終えれば，あなたは自分だけのサーバーを AWS 上に立ち上げ，自由に計算を走らせることができるようになるのである！
準備 ​
ハンズオンのソースコードは GitHub の handson/ec2-get-started に置いてある．
このハンズオンは，基本的に AWS の無料枠 の範囲内で実行することができる．
まずは，ハンズオンを実行するための環境を整える． これらの環境整備は，後のハンズオンでも前提となるものなので確実にミスなく行っていただきたい．
  • AWS Account: ハンズオンを実行するには個人の AWS アカウントが必要である． AWS アカウントの取得については ??? を参照のこと．
  • Python と Node.js: 本ハンズオンを実行するには，Python (3.6 以上)，Node.js (12.0 以上) がインストールされていなければならない．
  • AWS CLI: AWS CLI のインストールについては， ??? を参照． ここに記載されている認証鍵の設定も済ませておくこと．
  • AWS CDK: AWS CDK のインストールについては， ??? を参照．
  • ソースコードのダウンロード: 本ハンズオンで使用するプログラムのソースコードを，以下のコマンドを使って GitHub からダウンロードする．
shell
$ git clone https://github.com/tomomano/learn-aws-by-coding.git
$ git clone https://github.com/tomomano/learn-aws-by-coding.git
あるいは， https://github.com/tomomano/learn-aws-by-coding のページに行って，右上のダウンロードボタンからダウンロードすることもできる．
Docker を使用する場合
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもパッケージ済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
使用方法については ??? を参照のこと．
SSH ​
SSH (secure shell) は Unix 系のリモートサーバーに安全にアクセスするためのツールである． 本ハンズオンでは， SSH を使って仮想サーバーにアクセスする． SSH に慣れていない読者のため，簡単な説明をここで行おう．
SSH による通信はすべて暗号化されているので，機密情報をインターネットを介して安全に送受信することができる． 本ハンズオンで，リモートのサーバーにアクセスするための SSH クライアントがローカルマシンにインストールされている必要がある． SSH クライアントは Linux/Mac には標準搭載されている． Windows の場合は WSL をインストールすることで SSH クライアントを利用することを推奨する (??? を参照)．
SSH コマンドの基本的な使い方を次に示す． <host name> はアクセスする先のサーバーの IP アドレスや DNS によるホストネームが入る． <user name> は接続する先のユーザー名である．
shell
$ ssh <user name>@<host name>
$ ssh <user name>@<host name>
SSH は平文のパスワードによる認証を行うこともできるが，より強固なセキュリティを施すため，公開鍵暗号方式(Public Key Cryptography)による認証を行うことが強く推奨されており， EC2 はこの方法でしかアクセスを許していない． 公開鍵暗号方式の仕組みについては各自勉強してほしい． 本ハンズオンにおいて大事なことは，EC2 インスタンスが公開鍵(Public key)を保持し，クライアントとなるコンピュータ(読者自身のコンピュータ)が秘密鍵(Private key)を保持する，という点である． EC2 のインスタンスには秘密鍵を持ったコンピュータのみがアクセスすることができる．逆に言うと，秘密鍵が漏洩すると第三者もサーバーにアクセスできることになるので，秘密鍵は絶対に漏洩することのないよう注意して管理する．
SSH コマンドでは，ログインのために使用する秘密鍵ファイルを -i もしくは --identity_file のオプションで指定することができる． たとえば，次のように使う．
shell
$ ssh -i Ec2SecretKey.pem <user name>@<host name>
$ ssh -i Ec2SecretKey.pem <user name>@<host name>
アプリケーションの説明 ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#1で作製するアプリケーションのアーキテクチャ
このアプリケーションではまず，VPC (Virtual Private Cloud) を使ってプライベートな仮想ネットワーク環境を立ち上げている． その VPC の public subnet の内側に，EC2 (Elatic Compute Cloud) の仮想サーバーを配置する． さらに，セキュリティのため， Security Group による EC2 インスタンスへのアクセス制限を設定している． このようにして作成された仮想サーバーに，SSH を使ってアクセスし，簡単な計算を行う．
figure_title のようなアプリケーションを，CDK を使って構築する．
早速ではあるが，今回のハンズオンで使用するプログラムを見てみよう (handson/ec2-get-started/app.py)．
python
class MyFirstEc2(core.Stack):
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e}from"./chunks/ec2_keypair_console.fc89ef69.js";import{a as t,_ as c}from"./chunks/VPC.e1acca4d.js";import{_ as r,c as y,o as C,R as A}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Hands-on #1: 初めての EC2 インスタンスを起動する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-ec2.md","filePath":"development/aws/handson-ec2.md","lastUpdated":1684859551000}'),i={name:"development/aws/handson-ec2.md"},F=A('
Hands-on #1: 初めての EC2 インスタンスを起動する ​
ハンズオンの第一回では， CDK を使って EC2 のインスタンス(仮想サーバー)を作成し，SSH でサーバーにログインする，という演習を行う． このハンズオンを終えれば，あなたは自分だけのサーバーを AWS 上に立ち上げ，自由に計算を走らせることができるようになるのである！
準備 ​
ハンズオンのソースコードは GitHub の handson/ec2-get-started に置いてある．
このハンズオンは，基本的に AWS の無料枠 の範囲内で実行することができる．
まずは，ハンズオンを実行するための環境を整える． これらの環境整備は，後のハンズオンでも前提となるものなので確実にミスなく行っていただきたい．
  • AWS Account: ハンズオンを実行するには個人の AWS アカウントが必要である． AWS アカウントの取得については ??? を参照のこと．
  • Python と Node.js: 本ハンズオンを実行するには，Python (3.6 以上)，Node.js (12.0 以上) がインストールされていなければならない．
  • AWS CLI: AWS CLI のインストールについては， ??? を参照． ここに記載されている認証鍵の設定も済ませておくこと．
  • AWS CDK: AWS CDK のインストールについては， ??? を参照．
  • ソースコードのダウンロード: 本ハンズオンで使用するプログラムのソースコードを，以下のコマンドを使って GitHub からダウンロードする．
shell
$ git clone https://github.com/tomomano/learn-aws-by-coding.git
$ git clone https://github.com/tomomano/learn-aws-by-coding.git
あるいは， https://github.com/tomomano/learn-aws-by-coding のページに行って，右上のダウンロードボタンからダウンロードすることもできる．
Docker を使用する場合
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもパッケージ済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
使用方法については ??? を参照のこと．
SSH ​
SSH (secure shell) は Unix 系のリモートサーバーに安全にアクセスするためのツールである． 本ハンズオンでは， SSH を使って仮想サーバーにアクセスする． SSH に慣れていない読者のため，簡単な説明をここで行おう．
SSH による通信はすべて暗号化されているので，機密情報をインターネットを介して安全に送受信することができる． 本ハンズオンで，リモートのサーバーにアクセスするための SSH クライアントがローカルマシンにインストールされている必要がある． SSH クライアントは Linux/Mac には標準搭載されている． Windows の場合は WSL をインストールすることで SSH クライアントを利用することを推奨する (??? を参照)．
SSH コマンドの基本的な使い方を次に示す． <host name> はアクセスする先のサーバーの IP アドレスや DNS によるホストネームが入る． <user name> は接続する先のユーザー名である．
shell
$ ssh <user name>@<host name>
$ ssh <user name>@<host name>
SSH は平文のパスワードによる認証を行うこともできるが，より強固なセキュリティを施すため，公開鍵暗号方式(Public Key Cryptography)による認証を行うことが強く推奨されており， EC2 はこの方法でしかアクセスを許していない． 公開鍵暗号方式の仕組みについては各自勉強してほしい． 本ハンズオンにおいて大事なことは，EC2 インスタンスが公開鍵(Public key)を保持し，クライアントとなるコンピュータ(読者自身のコンピュータ)が秘密鍵(Private key)を保持する，という点である． EC2 のインスタンスには秘密鍵を持ったコンピュータのみがアクセスすることができる．逆に言うと，秘密鍵が漏洩すると第三者もサーバーにアクセスできることになるので，秘密鍵は絶対に漏洩することのないよう注意して管理する．
SSH コマンドでは，ログインのために使用する秘密鍵ファイルを -i もしくは --identity_file のオプションで指定することができる． たとえば，次のように使う．
shell
$ ssh -i Ec2SecretKey.pem <user name>@<host name>
$ ssh -i Ec2SecretKey.pem <user name>@<host name>
アプリケーションの説明 ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#1で作製するアプリケーションのアーキテクチャ
このアプリケーションではまず，VPC (Virtual Private Cloud) を使ってプライベートな仮想ネットワーク環境を立ち上げている． その VPC の public subnet の内側に，EC2 (Elatic Compute Cloud) の仮想サーバーを配置する． さらに，セキュリティのため， Security Group による EC2 インスタンスへのアクセス制限を設定している． このようにして作成された仮想サーバーに，SSH を使ってアクセスし，簡単な計算を行う．
figure_title のようなアプリケーションを，CDK を使って構築する．
早速ではあるが，今回のハンズオンで使用するプログラムを見てみよう (handson/ec2-get-started/app.py)．
python
class MyFirstEc2(core.Stack):
 
     def __init__(self, scope: core.App, name: str, key_name: str, **kwargs) -> None:
         super().__init__(scope, name, **kwargs)
diff --git a/assets/development_aws_handson-ec2.md.64c416d6.lean.js b/assets/development_aws_handson-ec2.md.d527a33c.lean.js
similarity index 90%
rename from assets/development_aws_handson-ec2.md.64c416d6.lean.js
rename to assets/development_aws_handson-ec2.md.d527a33c.lean.js
index ba674d42..ae97dcaa 100644
--- a/assets/development_aws_handson-ec2.md.64c416d6.lean.js
+++ b/assets/development_aws_handson-ec2.md.d527a33c.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e}from"./chunks/ec2_keypair_console.fc89ef69.js";import{a as t,_ as c}from"./chunks/VPC.e1acca4d.js";import{_ as r,c as y,o as C,R as A}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Hands-on #1: 初めての EC2 インスタンスを起動する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-ec2.md","filePath":"development/aws/handson-ec2.md","lastUpdated":1684858945000}'),i={name:"development/aws/handson-ec2.md"},F=A("",135),B=[F];function D(d,u,m,h,b,E){return C(),y("div",null,B)}const k=r(i,[["render",D]]);export{f as __pageData,k as default};
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e}from"./chunks/ec2_keypair_console.fc89ef69.js";import{a as t,_ as c}from"./chunks/VPC.e1acca4d.js";import{_ as r,c as y,o as C,R as A}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Hands-on #1: 初めての EC2 インスタンスを起動する","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-ec2.md","filePath":"development/aws/handson-ec2.md","lastUpdated":1684859551000}'),i={name:"development/aws/handson-ec2.md"},F=A("",135),B=[F];function D(d,u,m,h,b,E){return C(),y("div",null,B)}const k=r(i,[["render",D]]);export{f as __pageData,k as default};
diff --git a/assets/development_aws_handson-jupyter.md.c0838dc1.js b/assets/development_aws_handson-jupyter.md.887c5eff.js
similarity index 99%
rename from assets/development_aws_handson-jupyter.md.c0838dc1.js
rename to assets/development_aws_handson-jupyter.md.887c5eff.js
index dfbcbf6d..00d02489 100644
--- a/assets/development_aws_handson-jupyter.md.c0838dc1.js
+++ b/assets/development_aws_handson-jupyter.md.887c5eff.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as F,l as i,m as A,n as D,o as B,p as C}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as d,c as u,o as m,R as h}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"Hands-on #2: AWS でディープラーニングを実践","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-jupyter.md","filePath":"development/aws/handson-jupyter.md","lastUpdated":1684858945000}'),b={name:"development/aws/handson-jupyter.md"},f=h('
Hands-on #2: AWS でディープラーニングを実践 ​
準備 ​
ハンズオン第二回では， GPU を搭載した EC2 インスタンスを起動し，深層学習モデルの学習と推論を実行する演習を行う．
ハンズオンのソースコードは GitHub の handson/mnist に置いてある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． それ以外に必要な準備はない．
初期状態の AWS アカウントでは， GPU 搭載の G タイプのインスタンスの起動上限が 0 になっていることがある． これを確認するには， AWS コンソールから EC2 の画面を開き，左のメニューから Limits を選択する． その中の Running On-Demand All G instances という数字が G インスタンスの起動上限を表している．
もし，これが 0 になっていた場合は， AWS の自動申請フォームから上限緩和のリクエストを送る必要がある． 詳しくは 公式ドキュメンテーション "Amazon EC2 service quotas" を参照のこと．
このハンズオンは， g4dn.xlarge タイプの EC2 インスタンスを使うので，東京 (ap-northeast-1) リージョンでは 0.71 $/hour のコストが発生する．
AWS Educate Starter Account を使用している読者へ: 執筆時点においては， Starter Account には GPU 搭載型インスタンスを起動できないという制限が設けられている． したがって， Starter Account のユーザーはこのハンズオンを実行することはできない． 興味のある読者は，制限のない一般アカウントを自分自身で取得する必要があることに注意．
アプリケーションの説明 ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#2で作製するアプリケーションのアーキテクチャ
図の多くの部分が，第一回ハンズオンで作成したアプリケーションと共通していることに気がつくだろう． 少しの変更で，簡単にディープラーニングを走らせる環境を構築することができるのである！主な変更点は次の３点である．
  • GPU を搭載した g4dn.xlarge インスタンスタイプを使用
  • ディープラーニングに使うプログラムがあらかじめインストールされた DLAMI (後述) を使用
  • SSH にポートフォワーディングのオプションつけてサーバーに接続し，サーバーで起動している Jupyter Notebook (後述) を使ってプログラムを書いたり実行したりする
ハンズオンで使用するプログラムのコードをみてみよう handson/mnist/app.py)． コードは第一回目とほとんど共通である．変更点のみ解説を行う．
python
class Ec2ForDl(core.Stack):
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as F,l as i,m as A,n as D,o as B,p as C}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as d,c as u,o as m,R as h}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"Hands-on #2: AWS でディープラーニングを実践","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-jupyter.md","filePath":"development/aws/handson-jupyter.md","lastUpdated":1684859551000}'),b={name:"development/aws/handson-jupyter.md"},f=h('
Hands-on #2: AWS でディープラーニングを実践 ​
準備 ​
ハンズオン第二回では， GPU を搭載した EC2 インスタンスを起動し，深層学習モデルの学習と推論を実行する演習を行う．
ハンズオンのソースコードは GitHub の handson/mnist に置いてある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． それ以外に必要な準備はない．
初期状態の AWS アカウントでは， GPU 搭載の G タイプのインスタンスの起動上限が 0 になっていることがある． これを確認するには， AWS コンソールから EC2 の画面を開き，左のメニューから Limits を選択する． その中の Running On-Demand All G instances という数字が G インスタンスの起動上限を表している．
もし，これが 0 になっていた場合は， AWS の自動申請フォームから上限緩和のリクエストを送る必要がある． 詳しくは 公式ドキュメンテーション "Amazon EC2 service quotas" を参照のこと．
このハンズオンは， g4dn.xlarge タイプの EC2 インスタンスを使うので，東京 (ap-northeast-1) リージョンでは 0.71 $/hour のコストが発生する．
AWS Educate Starter Account を使用している読者へ: 執筆時点においては， Starter Account には GPU 搭載型インスタンスを起動できないという制限が設けられている． したがって， Starter Account のユーザーはこのハンズオンを実行することはできない． 興味のある読者は，制限のない一般アカウントを自分自身で取得する必要があることに注意．
アプリケーションの説明 ​
このハンズオンで作成するアプリケーションの概要を figure_title に示す．
ハンズオン#2で作製するアプリケーションのアーキテクチャ
図の多くの部分が，第一回ハンズオンで作成したアプリケーションと共通していることに気がつくだろう． 少しの変更で，簡単にディープラーニングを走らせる環境を構築することができるのである！主な変更点は次の３点である．
  • GPU を搭載した g4dn.xlarge インスタンスタイプを使用
  • ディープラーニングに使うプログラムがあらかじめインストールされた DLAMI (後述) を使用
  • SSH にポートフォワーディングのオプションつけてサーバーに接続し，サーバーで起動している Jupyter Notebook (後述) を使ってプログラムを書いたり実行したりする
ハンズオンで使用するプログラムのコードをみてみよう handson/mnist/app.py)． コードは第一回目とほとんど共通である．変更点のみ解説を行う．
python
class Ec2ForDl(core.Stack):
 
     def __init__(self, scope: core.App, name: str, key_name: str, **kwargs) -> None:
         super().__init__(scope, name, **kwargs)
diff --git a/assets/development_aws_handson-jupyter.md.c0838dc1.lean.js b/assets/development_aws_handson-jupyter.md.887c5eff.lean.js
similarity index 90%
rename from assets/development_aws_handson-jupyter.md.c0838dc1.lean.js
rename to assets/development_aws_handson-jupyter.md.887c5eff.lean.js
index ac8581af..b0e09175 100644
--- a/assets/development_aws_handson-jupyter.md.c0838dc1.lean.js
+++ b/assets/development_aws_handson-jupyter.md.887c5eff.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as F,l as i,m as A,n as D,o as B,p as C}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as d,c as u,o as m,R as h}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"Hands-on #2: AWS でディープラーニングを実践","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-jupyter.md","filePath":"development/aws/handson-jupyter.md","lastUpdated":1684858945000}'),b={name:"development/aws/handson-jupyter.md"},f=h("",155),g=[f];function E(v,_,k,q,S,P){return m(),u("div",null,g)}const T=d(b,[["render",E]]);export{w as __pageData,T as default};
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as F,l as i,m as A,n as D,o as B,p as C}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as d,c as u,o as m,R as h}from"./chunks/framework.a1bac067.js";const w=JSON.parse('{"title":"Hands-on #2: AWS でディープラーニングを実践","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-jupyter.md","filePath":"development/aws/handson-jupyter.md","lastUpdated":1684859551000}'),b={name:"development/aws/handson-jupyter.md"},f=h("",155),g=[f];function E(v,_,k,q,S,P){return m(),u("div",null,g)}const T=d(b,[["render",E]]);export{w as __pageData,T as default};
diff --git a/assets/development_aws_handson-qabot.md.2f7eac1b.js b/assets/development_aws_handson-qabot.md.b244b90e.js
similarity index 99%
rename from assets/development_aws_handson-qabot.md.2f7eac1b.js
rename to assets/development_aws_handson-qabot.md.b244b90e.js
index 4f279049..bd3c1a3f 100644
--- a/assets/development_aws_handson-qabot.md.2f7eac1b.js
+++ b/assets/development_aws_handson-qabot.md.b244b90e.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as i}from"./chunks/ask_many_output.0bb19110.js";import{_ as y,c as F,o as A,R as B}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"Hands-on #3: AWS で自動質問回答ボットを走らせる","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-qabot.md","filePath":"development/aws/handson-qabot.md","lastUpdated":1684858945000}'),D={name:"development/aws/handson-qabot.md"},C=B('
Hands-on #3: AWS で自動質問回答ボットを走らせる ​
ハンズオン第三回では， Docker と ECS を駆使した機械学習アプリケーションを実装しよう． 具体的には，深層学習による自然言語処理を行うことで，クライアントから与えられた文章題に対して回答を生成する，自動 Question & Answering ボットを作成しよう． ECS を利用することで，ジョブの数によって動的にインスタンスの数を制御し，並列にタスクを実行するようなシステムを構築しよう．
通常の機械学習のワークフローでは，モデルの訓練 ⇒ 推論 (データへの適用) が基本的な流れである． しかしながら， GPU 搭載型の EC2 クラスターを使ったモデルの訓練はやや難易度が高いため，次章 (???) で取り扱う． 本章は，クラウド上でのクラスターの構築・タスクの管理などの概念に慣れるため，よりシンプルな実装で実現できる Fargate クラスターを用いた推論計算の並列化を紹介する．
Fargate ​
ハンズオンに入っていく前に， Fargate という AWS の機能を知っておく必要がある (figure_title)．
Fargate のアイコン
ECS の概要を示した ??? をもう一度見てみよう． この図で， ECS の管理下にあるクラスターが示されているが，このクラスターの中で計算を行う実体としては二つの選択肢がある． EC2 あるいは Fargate のいずれかである． EC2 を用いた場合は，先の章 (???, ???) で説明したような流れでインスタンスが起動し，計算が実行される． しかし， EC2 を用いた計算機クラスターの作成・管理は技術的な難易度がやや高いので，次章 (???) で説明することにする．
Fargate とは， ECS での利用に特化して設計された，コンテナを使用した計算タスクを走らせるための仕組みである． 計算を走らせるという点では EC2 と役割は似ているが， Fargate は EC2 インスタンスのような物理的実体はもたない． 物理的実体をもたないというのは，たとえば SSH でログインすることは基本的に想定されていないし，なにかのソフトウェアをインストールしたりなどの概念も存在しない． Fargate ではすべての計算は Docker コンテナを介して行われる． すなわち， Fargate を利用するには，ユーザーは最初に所望の Docker イメージを指定しておき， Fargate は docker run のコマンドを使用することで計算タスクを実行する． Fargate を用いる利点は， Fargate を ECS のクラスターに指定すると，スケーリングなどの操作が簡単な設定・プログラムで構築できる点である．
Fargate では， EC2 と同様に CPU とメモリーのサイズを必要な分だけ指定できる． 執筆時点では， CPU は 0.25 - 4 コア， RAM は 0.5 - 30 GB の間で選択することができる (詳しくは 公式ドキュメンテーション "Amazon ECS on AWS Fargate" 参照)． クラスターのスケーリングが容易な分， Fargate では EC2 ほど大きな CPU コア・ RAM 容量を単一インスタンスに付与することができず，また GPU を利用することもできない．
以上が Fargate の概要であったが，くどくど言葉で説明してもなかなかピンとこないだろう． ここからは実際に手を動かしながら， ECS と Fargate を使った並列タスクの処理の仕方を学んでいこう．
厳密には， ECS に付与するクラスターには EC2 と Fargate のハイブリッドを使用することも可能である．
準備 ​
ハンズオンのソースコードは GitHub の handson/qa-bot にある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． また， Docker が自身のローカルマシンにインストール済みであることも必要である．
このハンズオンでは 1CPU/4GB RAM の Fargate インスタンスを使用する． 計算の実行には 0.025 $/hour のコストが発生することに注意．
Transformer を用いた question-answering プログラム ​
このハンズオンで開発する，自動質問回答システムをより具体的に定義しよう． 次のような文脈 (context) と質問 (question) が与えられた状況を想定する．
txt
context: Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). His work is also known for its influence on the philosophy of science. He is best known to the general public for his mass–energy equivalence formula E = mc2, which has been dubbed \\"the world's most famous equation\\". He received the 1921 Nobel Prize in Physics \\"for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect\\", a pivotal step in the development of quantum theory.
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as i}from"./chunks/ask_many_output.0bb19110.js";import{_ as y,c as F,o as A,R as B}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"Hands-on #3: AWS で自動質問回答ボットを走らせる","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-qabot.md","filePath":"development/aws/handson-qabot.md","lastUpdated":1684859551000}'),D={name:"development/aws/handson-qabot.md"},C=B('
Hands-on #3: AWS で自動質問回答ボットを走らせる ​
ハンズオン第三回では， Docker と ECS を駆使した機械学習アプリケーションを実装しよう． 具体的には，深層学習による自然言語処理を行うことで，クライアントから与えられた文章題に対して回答を生成する，自動 Question & Answering ボットを作成しよう． ECS を利用することで，ジョブの数によって動的にインスタンスの数を制御し，並列にタスクを実行するようなシステムを構築しよう．
通常の機械学習のワークフローでは，モデルの訓練 ⇒ 推論 (データへの適用) が基本的な流れである． しかしながら， GPU 搭載型の EC2 クラスターを使ったモデルの訓練はやや難易度が高いため，次章 (???) で取り扱う． 本章は，クラウド上でのクラスターの構築・タスクの管理などの概念に慣れるため，よりシンプルな実装で実現できる Fargate クラスターを用いた推論計算の並列化を紹介する．
Fargate ​
ハンズオンに入っていく前に， Fargate という AWS の機能を知っておく必要がある (figure_title)．
Fargate のアイコン
ECS の概要を示した ??? をもう一度見てみよう． この図で， ECS の管理下にあるクラスターが示されているが，このクラスターの中で計算を行う実体としては二つの選択肢がある． EC2 あるいは Fargate のいずれかである． EC2 を用いた場合は，先の章 (???, ???) で説明したような流れでインスタンスが起動し，計算が実行される． しかし， EC2 を用いた計算機クラスターの作成・管理は技術的な難易度がやや高いので，次章 (???) で説明することにする．
Fargate とは， ECS での利用に特化して設計された，コンテナを使用した計算タスクを走らせるための仕組みである． 計算を走らせるという点では EC2 と役割は似ているが， Fargate は EC2 インスタンスのような物理的実体はもたない． 物理的実体をもたないというのは，たとえば SSH でログインすることは基本的に想定されていないし，なにかのソフトウェアをインストールしたりなどの概念も存在しない． Fargate ではすべての計算は Docker コンテナを介して行われる． すなわち， Fargate を利用するには，ユーザーは最初に所望の Docker イメージを指定しておき， Fargate は docker run のコマンドを使用することで計算タスクを実行する． Fargate を用いる利点は， Fargate を ECS のクラスターに指定すると，スケーリングなどの操作が簡単な設定・プログラムで構築できる点である．
Fargate では， EC2 と同様に CPU とメモリーのサイズを必要な分だけ指定できる． 執筆時点では， CPU は 0.25 - 4 コア， RAM は 0.5 - 30 GB の間で選択することができる (詳しくは 公式ドキュメンテーション "Amazon ECS on AWS Fargate" 参照)． クラスターのスケーリングが容易な分， Fargate では EC2 ほど大きな CPU コア・ RAM 容量を単一インスタンスに付与することができず，また GPU を利用することもできない．
以上が Fargate の概要であったが，くどくど言葉で説明してもなかなかピンとこないだろう． ここからは実際に手を動かしながら， ECS と Fargate を使った並列タスクの処理の仕方を学んでいこう．
厳密には， ECS に付与するクラスターには EC2 と Fargate のハイブリッドを使用することも可能である．
準備 ​
ハンズオンのソースコードは GitHub の handson/qa-bot にある．
本ハンズオンの実行には，第一回ハンズオンで説明した準備 (???) が整っていることを前提とする． また， Docker が自身のローカルマシンにインストール済みであることも必要である．
このハンズオンでは 1CPU/4GB RAM の Fargate インスタンスを使用する． 計算の実行には 0.025 $/hour のコストが発生することに注意．
Transformer を用いた question-answering プログラム ​
このハンズオンで開発する，自動質問回答システムをより具体的に定義しよう． 次のような文脈 (context) と質問 (question) が与えられた状況を想定する．
txt
context: Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). His work is also known for its influence on the philosophy of science. He is best known to the general public for his mass–energy equivalence formula E = mc2, which has been dubbed \\"the world's most famous equation\\". He received the 1921 Nobel Prize in Physics \\"for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect\\", a pivotal step in the development of quantum theory.
 
 question: In what year did Einstein win the Nobel prize?
context: Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). His work is also known for its influence on the philosophy of science. He is best known to the general public for his mass–energy equivalence formula E = mc2, which has been dubbed \\"the world's most famous equation\\". He received the 1921 Nobel Prize in Physics \\"for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect\\", a pivotal step in the development of quantum theory.
 
diff --git a/assets/development_aws_handson-qabot.md.2f7eac1b.lean.js b/assets/development_aws_handson-qabot.md.b244b90e.lean.js
similarity index 90%
rename from assets/development_aws_handson-qabot.md.2f7eac1b.lean.js
rename to assets/development_aws_handson-qabot.md.b244b90e.lean.js
index 88344c2f..d313be58 100644
--- a/assets/development_aws_handson-qabot.md.2f7eac1b.lean.js
+++ b/assets/development_aws_handson-qabot.md.b244b90e.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as i}from"./chunks/ask_many_output.0bb19110.js";import{_ as y,c as F,o as A,R as B}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"Hands-on #3: AWS で自動質問回答ボットを走らせる","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-qabot.md","filePath":"development/aws/handson-qabot.md","lastUpdated":1684858945000}'),D={name:"development/aws/handson-qabot.md"},C=B("",103),u=[C];function d(m,h,b,f,g,E){return A(),F("div",null,u)}const k=y(D,[["render",d]]);export{v as __pageData,k as default};
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as i}from"./chunks/ask_many_output.0bb19110.js";import{_ as y,c as F,o as A,R as B}from"./chunks/framework.a1bac067.js";const v=JSON.parse('{"title":"Hands-on #3: AWS で自動質問回答ボットを走らせる","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-qabot.md","filePath":"development/aws/handson-qabot.md","lastUpdated":1684859551000}'),D={name:"development/aws/handson-qabot.md"},C=B("",103),u=[C];function d(m,h,b,f,g,E){return A(),F("div",null,u)}const k=y(D,[["render",d]]);export{v as __pageData,k as default};
diff --git a/assets/development_aws_handson-serverless.md.38596c90.js b/assets/development_aws_handson-serverless.md.a47ced9b.js
similarity index 99%
rename from assets/development_aws_handson-serverless.md.38596c90.js
rename to assets/development_aws_handson-serverless.md.a47ced9b.js
index f0d7c782..2233bb7a 100644
--- a/assets/development_aws_handson-serverless.md.38596c90.js
+++ b/assets/development_aws_handson-serverless.md.a47ced9b.js
@@ -1,4 +1,4 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as i,l as F}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as A,c as B,o as d,R as D}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Hands-on #5: サーバーレス入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-serverless.md","filePath":"development/aws/handson-serverless.md","lastUpdated":1684858945000}'),C={name:"development/aws/handson-serverless.md"},m=D('
Hands-on #5: サーバーレス入門 ​
前章ではサーバーレスアーキテクチャの概要の説明を行った． 本章では，ハンズオン形式でサーバーレスクラウドを実際に動かしながら，具体的な使用方法を学んでいこう． 今回のハンズオンでは Lambda, S3, DynamoDB の三つのサーバーレスクラウドの構成要素に触れていく． それぞれについて，短いチュートリアルを用意してある．
Lambda ハンズオン ​
まずは， Lambda を実際に動かしてみよう． ハンズオンのソースコードは GitHub の handson/serverless/lambda に置いてある．
このハンズオンで使用するアプリケーションのスケッチを figure_title に示す． STEP 1 では，AWS CDK を使用して Python で書かれたコードを Lambda に登録する． 続いて STEP 2 では， Invoke API を使用して，同時にいくつもの Lambda を起動し，並列な計算を行う． Lambda のワークフローを体験する目的で最小限の設定である．
Lambda チュートリアルの概要
このハンズオンは，基本的に AWS Lambda の無料枠 の範囲内で実行することができる．
app.py にデプロイするプログラムが書かれている． 中身を見てみよう．
python
#
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as i,l as F}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as A,c as B,o as d,R as D}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Hands-on #5: サーバーレス入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-serverless.md","filePath":"development/aws/handson-serverless.md","lastUpdated":1684859551000}'),C={name:"development/aws/handson-serverless.md"},m=D('
Hands-on #5: サーバーレス入門 ​
前章ではサーバーレスアーキテクチャの概要の説明を行った． 本章では，ハンズオン形式でサーバーレスクラウドを実際に動かしながら，具体的な使用方法を学んでいこう． 今回のハンズオンでは Lambda, S3, DynamoDB の三つのサーバーレスクラウドの構成要素に触れていく． それぞれについて，短いチュートリアルを用意してある．
Lambda ハンズオン ​
まずは， Lambda を実際に動かしてみよう． ハンズオンのソースコードは GitHub の handson/serverless/lambda に置いてある．
このハンズオンで使用するアプリケーションのスケッチを figure_title に示す． STEP 1 では，AWS CDK を使用して Python で書かれたコードを Lambda に登録する． 続いて STEP 2 では， Invoke API を使用して，同時にいくつもの Lambda を起動し，並列な計算を行う． Lambda のワークフローを体験する目的で最小限の設定である．
Lambda チュートリアルの概要
このハンズオンは，基本的に AWS Lambda の無料枠 の範囲内で実行することができる．
app.py にデプロイするプログラムが書かれている． 中身を見てみよう．
python
#
 FUNC = """
 import time
 from random import choice, randint
diff --git a/assets/development_aws_handson-serverless.md.38596c90.lean.js b/assets/development_aws_handson-serverless.md.a47ced9b.lean.js
similarity index 90%
rename from assets/development_aws_handson-serverless.md.38596c90.lean.js
rename to assets/development_aws_handson-serverless.md.a47ced9b.lean.js
index 340b182e..e1671700 100644
--- a/assets/development_aws_handson-serverless.md.38596c90.lean.js
+++ b/assets/development_aws_handson-serverless.md.a47ced9b.lean.js
@@ -1 +1 @@
-import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as i,l as F}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as A,c as B,o as d,R as D}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Hands-on #5: サーバーレス入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-serverless.md","filePath":"development/aws/handson-serverless.md","lastUpdated":1684858945000}'),C={name:"development/aws/handson-serverless.md"},m=D("",125),b=[m];function u(h,_,f,v,g,E){return d(),B("div",null,b)}const w=A(C,[["render",u]]);export{q as __pageData,w as default};
+import{_ as s,a,b as n,c as l,d as p,e as o,f as e,g as t,h as r,i as c,j as y,k as i,l as F}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as A,c as B,o as d,R as D}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Hands-on #5: サーバーレス入門","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/handson-serverless.md","filePath":"development/aws/handson-serverless.md","lastUpdated":1684859551000}'),C={name:"development/aws/handson-serverless.md"},m=D("",125),b=[m];function u(h,_,f,v,g,E){return d(),B("div",null,b)}const w=A(C,[["render",u]]);export{q as __pageData,w as default};
diff --git a/assets/development_aws_introduction.md.222bee89.js b/assets/development_aws_introduction.md.a28e3444.js
similarity index 99%
rename from assets/development_aws_introduction.md.222bee89.js
rename to assets/development_aws_introduction.md.a28e3444.js
index c4477adb..4a1bea1a 100644
--- a/assets/development_aws_introduction.md.222bee89.js
+++ b/assets/development_aws_introduction.md.a28e3444.js
@@ -1 +1 @@
-import{_ as a}from"./chunks/earth_from_earth.8c108f53.js";import{_ as e,c as t,o,R as r}from"./chunks/framework.a1bac067.js";const S=JSON.parse('{"title":"はじめに","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/introduction.md","filePath":"development/aws/introduction.md","lastUpdated":1684858945000}'),s={name:"development/aws/introduction.md"},l=r('
はじめに ​
本書の目的・内容 ​
本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．
本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．
とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．
本書は以下のような三部構成になっている．
本書の構成
テーマハンズオン
第一部 (1章-4章)
クラウドの基礎
  • AWSに自分のサーバーを立ち上げる
第二部 (5章-9章)
クラウドを活用した機械学習
  • AWS と Jupyter を使って始めるディープラーニング
  • スケーラブルな自動質問回答ボットを作る
  • 並列化されたハイパーパラメータサーチの実装
第三部 (10章-13章)
サーバーレスアーキテクチャ入門
  • Lambda, DynamoDB, S3 の演習
  • 俳句を投稿する SNS "Bashoutter" を作る
第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．
第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．
第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．
これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．
本書のフィロソフィー ​
本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．
どういうことか？
ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．
ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．
そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．
決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．
宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)
AWS アカウント ​
本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (???) に簡単に記載したので，必要に応じて参照していただきたい．
AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．
また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．
環境構築 ​
本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (???) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．
  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (??? 参照)．
  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については ??? を参照のこと．
  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は ??? 参照のこと．
  • Node.js: version 12.0 以上 をインストールする．
  • AWS CLI: Version 2 をインストールする． インストール手順については ??? 参照のこと．
  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については ??? 参照のこと．
  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については ??? 参照のこと．
ハンズオン実行用の Docker Image ​
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
次のコマンドで起動する．
shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc
この Docker image の使い方や詳細は ??? に記載している．
前提知識 ​
本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．
  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．
  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．
講義に関連する資料 ​
ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．
https://github.com/tomomano/learn-aws-by-coding
本書で使用するノーテーションなど ​
  • コードやシェルのコマンドは monospace letter で記述する．
  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．
また，以下のような形式で注意やチップスを提供する．
追加のコメントなどを記す．
発展的な議論やアイディアなどを紹介する．
陥りやすいミスなどの注意事項を述べる．
絶対に犯してはならないミスを指摘する．
',44),n=[l];function p(i,c,d,h,u,m){return o(),t("div",null,n)}const b=e(s,[["render",p]]);export{S as __pageData,b as default};
+import{_ as a}from"./chunks/earth_from_earth.8c108f53.js";import{_ as e,c as t,o,R as r}from"./chunks/framework.a1bac067.js";const S=JSON.parse('{"title":"はじめに","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/introduction.md","filePath":"development/aws/introduction.md","lastUpdated":1684859551000}'),s={name:"development/aws/introduction.md"},l=r('
はじめに ​
本書の目的・内容 ​
本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．
本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．
とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．
本書は以下のような三部構成になっている．
本書の構成
テーマハンズオン
第一部 (1章-4章)
クラウドの基礎
  • AWSに自分のサーバーを立ち上げる
第二部 (5章-9章)
クラウドを活用した機械学習
  • AWS と Jupyter を使って始めるディープラーニング
  • スケーラブルな自動質問回答ボットを作る
  • 並列化されたハイパーパラメータサーチの実装
第三部 (10章-13章)
サーバーレスアーキテクチャ入門
  • Lambda, DynamoDB, S3 の演習
  • 俳句を投稿する SNS "Bashoutter" を作る
第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．
第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．
第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．
これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．
本書のフィロソフィー ​
本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．
どういうことか？
ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．
ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．
そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．
決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．
宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)
AWS アカウント ​
本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (???) に簡単に記載したので，必要に応じて参照していただきたい．
AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．
また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．
環境構築 ​
本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (???) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．
  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (??? 参照)．
  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については ??? を参照のこと．
  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は ??? 参照のこと．
  • Node.js: version 12.0 以上 をインストールする．
  • AWS CLI: Version 2 をインストールする． インストール手順については ??? 参照のこと．
  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については ??? 参照のこと．
  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については ??? 参照のこと．
ハンズオン実行用の Docker Image ​
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
次のコマンドで起動する．
shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc
この Docker image の使い方や詳細は ??? に記載している．
前提知識 ​
本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．
  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．
  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．
講義に関連する資料 ​
ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．
https://github.com/tomomano/learn-aws-by-coding
本書で使用するノーテーションなど ​
  • コードやシェルのコマンドは monospace letter で記述する．
  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．
また，以下のような形式で注意やチップスを提供する．
追加のコメントなどを記す．
発展的な議論やアイディアなどを紹介する．
陥りやすいミスなどの注意事項を述べる．
絶対に犯してはならないミスを指摘する．
',44),n=[l];function p(i,c,d,h,u,m){return o(),t("div",null,n)}const b=e(s,[["render",p]]);export{S as __pageData,b as default};
diff --git a/assets/development_aws_introduction.md.222bee89.lean.js b/assets/development_aws_introduction.md.a28e3444.lean.js
similarity index 87%
rename from assets/development_aws_introduction.md.222bee89.lean.js
rename to assets/development_aws_introduction.md.a28e3444.lean.js
index 5bc44fe5..31b01194 100644
--- a/assets/development_aws_introduction.md.222bee89.lean.js
+++ b/assets/development_aws_introduction.md.a28e3444.lean.js
@@ -1 +1 @@
-import{_ as a}from"./chunks/earth_from_earth.8c108f53.js";import{_ as e,c as t,o,R as r}from"./chunks/framework.a1bac067.js";const S=JSON.parse('{"title":"はじめに","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/introduction.md","filePath":"development/aws/introduction.md","lastUpdated":1684858945000}'),s={name:"development/aws/introduction.md"},l=r("",44),n=[l];function p(i,c,d,h,u,m){return o(),t("div",null,n)}const b=e(s,[["render",p]]);export{S as __pageData,b as default};
+import{_ as a}from"./chunks/earth_from_earth.8c108f53.js";import{_ as e,c as t,o,R as r}from"./chunks/framework.a1bac067.js";const S=JSON.parse('{"title":"はじめに","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/introduction.md","filePath":"development/aws/introduction.md","lastUpdated":1684859551000}'),s={name:"development/aws/introduction.md"},l=r("",44),n=[l];function p(i,c,d,h,u,m){return o(),t("div",null,n)}const b=e(s,[["render",p]]);export{S as __pageData,b as default};
diff --git a/assets/development_aws_license.md.48a0d402.js b/assets/development_aws_license.md.3681aae7.js
similarity index 94%
rename from assets/development_aws_license.md.48a0d402.js
rename to assets/development_aws_license.md.3681aae7.js
index 6422f12a..60f7fe62 100644
--- a/assets/development_aws_license.md.48a0d402.js
+++ b/assets/development_aws_license.md.3681aae7.js
@@ -1 +1 @@
-import{_ as s,c as a,o as n,z as e,a as t}from"./chunks/framework.a1bac067.js";const c="/assets/cc_by_nc_nd.c97a96cd.png",x=JSON.parse('{"title":"ライセンス","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/license.md","filePath":"development/aws/license.md","lastUpdated":1684858945000}'),o={name:"development/aws/license.md"},r=e("h1",{id:"ライセンス",tabindex:"-1"},[t("ライセンス "),e("a",{class:"header-anchor",href:"#ライセンス","aria-label":'Permalink to "ライセンス"'},"​")],-1),l=e("p",null,[t("本教科書およびハンズオンのソースコードは "),e("a",{href:"https://creativecommons.org/licenses/by-nc-nd/4.0/",target:"_blank",rel:"noreferrer"},"CC BY-NC-ND 4.0"),t(" に従うライセンスで公開しています．")],-1),_=e("p",null,"教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．",-1),d=e("p",null,[e("img",{src:c,alt:"cc_by_nc_nd"})],-1),i=[r,l,_,d];function p(m,h,f,u,b,v){return n(),a("div",null,i)}const N=s(o,[["render",p]]);export{x as __pageData,N as default};
+import{_ as s,c as a,o as n,z as e,a as t}from"./chunks/framework.a1bac067.js";const c="/assets/cc_by_nc_nd.c97a96cd.png",x=JSON.parse('{"title":"ライセンス","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/license.md","filePath":"development/aws/license.md","lastUpdated":1684859551000}'),o={name:"development/aws/license.md"},r=e("h1",{id:"ライセンス",tabindex:"-1"},[t("ライセンス "),e("a",{class:"header-anchor",href:"#ライセンス","aria-label":'Permalink to "ライセンス"'},"​")],-1),l=e("p",null,[t("本教科書およびハンズオンのソースコードは "),e("a",{href:"https://creativecommons.org/licenses/by-nc-nd/4.0/",target:"_blank",rel:"noreferrer"},"CC BY-NC-ND 4.0"),t(" に従うライセンスで公開しています．")],-1),_=e("p",null,"教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．",-1),d=e("p",null,[e("img",{src:c,alt:"cc_by_nc_nd"})],-1),i=[r,l,_,d];function p(m,h,f,u,b,v){return n(),a("div",null,i)}const N=s(o,[["render",p]]);export{x as __pageData,N as default};
diff --git a/assets/development_aws_license.md.48a0d402.lean.js b/assets/development_aws_license.md.3681aae7.lean.js
similarity index 94%
rename from assets/development_aws_license.md.48a0d402.lean.js
rename to assets/development_aws_license.md.3681aae7.lean.js
index 6422f12a..60f7fe62 100644
--- a/assets/development_aws_license.md.48a0d402.lean.js
+++ b/assets/development_aws_license.md.3681aae7.lean.js
@@ -1 +1 @@
-import{_ as s,c as a,o as n,z as e,a as t}from"./chunks/framework.a1bac067.js";const c="/assets/cc_by_nc_nd.c97a96cd.png",x=JSON.parse('{"title":"ライセンス","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/license.md","filePath":"development/aws/license.md","lastUpdated":1684858945000}'),o={name:"development/aws/license.md"},r=e("h1",{id:"ライセンス",tabindex:"-1"},[t("ライセンス "),e("a",{class:"header-anchor",href:"#ライセンス","aria-label":'Permalink to "ライセンス"'},"​")],-1),l=e("p",null,[t("本教科書およびハンズオンのソースコードは "),e("a",{href:"https://creativecommons.org/licenses/by-nc-nd/4.0/",target:"_blank",rel:"noreferrer"},"CC BY-NC-ND 4.0"),t(" に従うライセンスで公開しています．")],-1),_=e("p",null,"教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．",-1),d=e("p",null,[e("img",{src:c,alt:"cc_by_nc_nd"})],-1),i=[r,l,_,d];function p(m,h,f,u,b,v){return n(),a("div",null,i)}const N=s(o,[["render",p]]);export{x as __pageData,N as default};
+import{_ as s,c as a,o as n,z as e,a as t}from"./chunks/framework.a1bac067.js";const c="/assets/cc_by_nc_nd.c97a96cd.png",x=JSON.parse('{"title":"ライセンス","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/license.md","filePath":"development/aws/license.md","lastUpdated":1684859551000}'),o={name:"development/aws/license.md"},r=e("h1",{id:"ライセンス",tabindex:"-1"},[t("ライセンス "),e("a",{class:"header-anchor",href:"#ライセンス","aria-label":'Permalink to "ライセンス"'},"​")],-1),l=e("p",null,[t("本教科書およびハンズオンのソースコードは "),e("a",{href:"https://creativecommons.org/licenses/by-nc-nd/4.0/",target:"_blank",rel:"noreferrer"},"CC BY-NC-ND 4.0"),t(" に従うライセンスで公開しています．")],-1),_=e("p",null,"教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．",-1),d=e("p",null,[e("img",{src:c,alt:"cc_by_nc_nd"})],-1),i=[r,l,_,d];function p(m,h,f,u,b,v){return n(),a("div",null,i)}const N=s(o,[["render",p]]);export{x as __pageData,N as default};
diff --git a/assets/development_aws_main.md.dd2f6ba8.js b/assets/development_aws_main.md.cc9bca88.js
similarity index 99%
rename from assets/development_aws_main.md.dd2f6ba8.js
rename to assets/development_aws_main.md.cc9bca88.js
index e0fb7b73..9df06621 100644
--- a/assets/development_aws_main.md.dd2f6ba8.js
+++ b/assets/development_aws_main.md.cc9bca88.js
@@ -1,4 +1,4 @@
-import{_ as n}from"./chunks/earth_from_earth.8c108f53.js";import{_ as l,a as p,b as o,c as e}from"./chunks/terminal.94539704.js";import{_ as t,a as r,b as c,c as y,d as i,e as F,f as A,g as B,h as D,i as C,j as d}from"./chunks/iac.2263bc12.js";import{_ as s,a}from"./chunks/VPC.e1acca4d.js";import{_ as u,a as m,b,c as h,d as g,e as f,f as E}from"./chunks/ec2_keypair_console.fc89ef69.js";import{_,a as v}from"./chunks/cnn.a8836fd9.js";import{_ as k,a as q,b as S,c as w,d as P,e as I,f as x,g as T,h as W,i as $,j as L,k as G,l as N,m as U,n as R,o as X,p as M}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as H,a as z,b as O,c as j,d as V,e as K,f as Y,g as J}from"./chunks/ecs.73d77e6a.js";import{_ as Q,a as Z,b as ss,c as as,d as ns,e as ls,f as ps,g as os,h as es,i as ts,j as rs}from"./chunks/ask_many_output.0bb19110.js";import{_ as cs,a as ys,b as is,c as Fs,d as As,e as Bs,f as Ds,g as Cs,h as ds,i as us,j as ms,k as bs,l as hs,m as gs,n as fs,o as Es,p as _s,q as vs,r as ks,s as qs}from"./chunks/cloud_development.e43e5d4a.js";import{_ as Ss,a as ws}from"./chunks/rest_api.602a6e96.js";import{_ as Ps,a as Is,b as xs,c as Ts,d as Ws}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as $s,a as Ls,b as Gs,c as Ns,d as Us,e as Rs,f as Xs,g as Ms,h as Hs,i as zs,j as Os,k as js,l as Vs}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as Ks,a as Ys,b as Js,c as Qs,d as Zs,e as sa,f as aa,g as na,h as la}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as pa,a as oa,b as ea,c as ta,d as ra,e as ca,f as ya,g as ia,h as Fa,i as Aa,j as Ba,k as Da,l as Ca,m as da,n as ua,o as ma}from"./chunks/venv_shell.dc2530bc.js";import{_ as ba,c as ha,o as ga,R as fa}from"./chunks/framework.a1bac067.js";const Ya=JSON.parse('{"title":"はじめに!","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/main.md","filePath":"development/aws/main.md","lastUpdated":1684858945000}'),Ea={name:"development/aws/main.md"},_a=fa('
はじめに! ​
本書の目的・内容 ​
本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．
本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．
とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．
本書は以下のような三部構成になっている．
本書の構成
テーマハンズオン
第一部 (1章-4章)
クラウドの基礎
  • AWSに自分のサーバーを立ち上げる
第二部 (5章-9章)
クラウドを活用した機械学習
  • AWS と Jupyter を使って始めるディープラーニング
  • スケーラブルな自動質問回答ボットを作る
  • 並列化されたハイパーパラメータサーチの実装
第三部 (10章-13章)
サーバーレスアーキテクチャ入門
  • Lambda, DynamoDB, S3 の演習
  • 俳句を投稿する SNS "Bashoutter" を作る
第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．
第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．
第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．
これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．
本書のフィロソフィー ​
本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．
どういうことか？
ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．
ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．
そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．
決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．
宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)
AWS アカウント ​
本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (AWS アカウントの取得) に簡単に記載したので，必要に応じて参照していただきたい．
AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．
また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．
環境構築 ​
本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (Appendix: 環境構築) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．
  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (WSL のインストール 参照)．
  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については Docker のインストール を参照のこと．
  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は Python クイックガイド 参照のこと．
  • Node.js: version 12.0 以上 をインストールする．
  • AWS CLI: Version 2 をインストールする． インストール手順については AWS CLI のインストール 参照のこと．
  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については AWS CDK のインストール 参照のこと．
  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については AWS CLI のインストール 参照のこと．
ハンズオン実行用の Docker Image ​
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
次のコマンドで起動する．
shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc
この Docker image の使い方や詳細は ハンズオン実行用の Docker image の使い方 に記載している．
前提知識 ​
本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．
  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．
  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．
講義に関連する資料 ​
ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．
https://github.com/tomomano/learn-aws-by-coding
本書で使用するノーテーションなど ​
  • コードやシェルのコマンドは monospace letter で記述する．
  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．
また，以下のような形式で注意やチップスを提供する．
追加のコメントなどを記す．
発展的な議論やアイディアなどを紹介する．
陥りやすいミスなどの注意事項を述べる．
絶対に犯してはならないミスを指摘する．
クラウド概論 ​
クラウドとは？ ​
Cloud
クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．
学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．
The NIST Definition of Cloud Computing
これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．
  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．
  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．
  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．
  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．
  • Measured service 計算資源の利用量を計測・監視することができる．
…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．
個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．
クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．
このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．
著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．
クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．
先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．
  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．
  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．
  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．
本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．
SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (Hands-on #6: Bashoutter) などの演習を提供する．
なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については Hands-on #5: サーバーレス入門 などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．
最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．
一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．
第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．
本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．
なぜクラウドを使うのか？ ​
上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．
  1. 自由にサーバーのサイズをスケールできる
    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない
    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0
    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．
    クラウドと自前計算機環境の経済的コストの比較
とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．
ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．
  1. クラウドは賢く使わないといけない
    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ
    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ
    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．
Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？
Terminal
この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．
時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．
最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．
AWS 入門 ​
AWS とは？ ​
本書では，クラウドの実践を行うプラットフォームとして， AWS を用いる． 実践にあたって，最低限必要な AWS の知識を本章では解説しよう．
AWS (Amazon Web Services) は Amazon 社が提供する総合的なクラウドプラットフォームである． AWS は Amazon 社が持つ膨大な計算リソースを貸し出すクラウドサービスとして，2006 年に誕生した． 2021 年では，クラウドプロバイダーとして最大のマーケットシェア (約 32%) を保持している (参照)． Netflix や Slack をはじめとした多くのウェブ関連のサービスで，一部または全てのサーバーリソースが AWS から提供されているとのことである． よって，知らないうちに AWS の恩恵にあずかっている人も少なくないはずだ．
最大のシェアをもつだけに，機能・サービスの幅広さはほかのクラウドプラットフォームと比べ抜きんでている． また，利用者数が多いことを反映して，公式あるいはサードパーティによる技術紹介記事が数多くウェブ上に存在しているだけでなく，ライブラリのユーザーコミュニティも大きく問題解決が捗るのも魅力の一つだ． 初期のころウェブビジネスを行う企業がユーザーの大半を占めていたが，最近は大学などでの科学研究用途としても頻繁に用いられるようになってきている．
AWS の機能・サービス ​
figure_title は，執筆時点において AWS で提供されている主要な機能・サービスの一覧である．
AWSで提供されている主要なサービス一覧
計算，ストレージ，データベース，ネットワーク，セキュリティなど，クラウドの構築に必要な様々な要素が独立したコンポーネントとして提供されている． 基本的に，これらを組み合わせることで一つのクラウドシステムができあがる．
また，機械学習・音声認識・AR/VR など，特定のアプリケーションにパッケージ済みのサービスも提供されている． これらを合計すると全部で 170 個以上のサービスが提供されているとのことである (参照)．
AWS の初心者が陥りがちなのは，大量のサービスの数に圧倒され，どこから手をつけたらよいのかわからなくなる，という状況である． たくさんのサービスの中から，どのサービスをどの順番で学んでいったらいいのか，その道筋すら明らかでなく，大きな参入障壁となっていることは間違いない． だが実のところ， AWS の基本的な構成要素はそのうちの数個のみに限られる． 基本要素となる機能の使い方を知れば， AWS のおおよそのリソースを使いこなすことが可能になる． ほかの機能の多くは，基本の要素を組み合わせて特定のアプリケーションに特化したパッケージとして AWS が用意したものである． そのポイントを認知することが， AWS の学習の最初のステップである．
ここでは， AWS 上でクラウドシステムを構築するときの基本となる構成要素を列挙する． これらは後のハンズオンで実際にプログラムを書きながら体験する． 現時点では，名前だけでも頭の片隅に記憶してもらえればよい．
計算 ​
S3 EC2 (Elastic Compute Cloud) 様々なスペックの仮想マシンを作成し，計算を実行することができる． クラウドの最も基本となる構成要素である． Hands-on #1: 初めての EC2 インスタンスを起動する, Hands-on #2: AWS でディープラーニングを実践, Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する で詳しく触れる．
S3 Lambda Function as a Service (FaaS) とよばれる，小さな計算をサーバーなしで実行するためのサービス． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
ストレージ ​
S3 EBS (Elastic Block Store) EC2 に付与することのできる仮想データドライブ． いわゆる"普通の"(一般的な OS で使われている)ファイルシステムを思い浮かべてくれたらよい．
S3 S3 (Simple Storage Service) Object Storage とよばれる，API を使ってデータの読み書きを行う，いうなれば”クラウド・ネイティブ”なデータの格納システムである． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
データベース ​
S3 DynamoDB NoSQL 型のデータベースサービス (知っている人は mongoDB などを思い浮かべたらよい)． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
ネットワーク ​
S3 VPC(Virtual Private Cloud) AWS 上に仮想ネットワーク環境を作成し，仮想サーバー間の接続を定義したり，外部からのアクセスなどを管理する． EC2 は VPC の内部に配置されなければならない．
API Gateway S3 API のエンドポイントとバックエンドのサービス (Lambda など) を接続する際に用いる，リバースプロキシとしての役割を担う． Hands-on #6: Bashoutter で詳しく解説する．
Region と Availability Zone ​
AWS を使用する際に知っておかなければならない重要な概念として， リージョン (Region) と Availability Zone (AZ) がある (figure_title)． 以下ではこの概念について簡単に記述する．
AWSにおける Region と Availability Zones
リージョン (Region) とは，おおまかに言うとデータセンターの所在地のことである． 執筆時点において， AWS は世界の 25 の国と地域でデータセンターを所有している． figure_title は執筆時点で利用できるリージョンの世界地図を示している． 日本では東京と大阪にデータセンターがある． 各リージョンには固有の ID がついており，例えば東京は ap-northeast-1, 米国オハイオ州は us-east-2，などと定義されている．
Regions in AWS(出典: https://aws.amazon.com/about-aws/global-infrastructure/)
AWS コンソールにログインすると，画面右上のメニューバーでリージョンを選択することができる(figure_title, 赤丸で囲った箇所)． EC2, S3 などの AWS のリソースは，リージョンごとに完全に独立である． したがって，リソースを新たにデプロイする際，あるいはデプロイ済みのリソースを閲覧する際は，コンソールのリージョンが正しく設定されているか，確認する必要がある． ウェブビジネスを展開する場合などは，世界の各地にクラウドを展開する必要があるが，個人的な研究用途として用いる場合は，最寄りのリージョン (i.e. 東京) を使えば基本的に問題ない．
AWSコンソールでリージョンを選択
Avaialibity Zone (AZ) とは，リージョン内で地理的に隔離されたデータセンターのことである． それぞれのリージョンは 2 個以上の AZ を有しており，もし一つの AZ で火災や停電などが起きた場合でも，ほかの AZ がその障害をカバーすることができる． また， AZ 間は高速な AWS 専用ネットワーク回線で結ばれているため， AZ 間のデータ転送は極めて早い． AZ は，ビジネスなどでサーバーダウンが許容されない場合などに注意すべき概念であり，個人的な用途で使う限りにおいてはあまり深く考慮する必要はない．言葉の意味だけ知っておけば十分である．
AWS を使用する際，どこのリージョンを指定するのがよいのだろうか？ インターネットの接続速度の観点からは，地理的に一番近いリージョンを使用するのが一般的によいだろう． 一方， EC2 の利用料などはリージョンごとに価格設定が若干 (10-20%程度) 異なる． したがって，自分が最も頻繁に利用するサービスの価格が最も安く設定されているリージョンを選択する，というのも重要な視点である． また，いくつかのサービスは，特定のリージョンで利用できない場合もある． これらのポイントから総合的に判断して使用するリージョンを決めると良い．
AWS Educate を利用している読者へ
執筆時点において，AWS Educate による Starter Account を使用している場合は us-east-1 region のみ利用できる (参照)．
AWS でのクラウド開発 ​
AWS のクラウドの全体像がわかってきたところで，次のトピックとして，どのようにして AWS 上にクラウドの開発を行い，展開していくかについての概略を解説しよう．
AWS のリソースを追加・編集・削除するなどの操作を実行するには，コンソールを用いる方法と，API を用いる方法の，二つの経路がある．
コンソール画面からリソースを操作する ​
AWS のアカウントにログインすると，まず最初に表示されるのがAWS コンソールである (figure_title)．
AWSマネージメントコンソール画面
コンソールを使うことで， EC2 のインスタンスを立ち上げたり，S3 のデータを追加・削除したり，ログを閲覧したりなど，AWS 上のあらゆるリソースの操作を GUI (Graphical User Interface) を通して実行することができる． 初めて触る機能をポチポチと試したり，デバッグを行うときなどにとても便利である．
コンソールはさらっと機能を試したり，開発中のクラウドのデバッグをするときには便利なのであるが，実際にクラウドの開発をする場面でこれを直接いじることはあまりない． むしろ，次に紹介する API を使用して，プログラムとしてクラウドのリソースを記述することで開発を行うのが一般的である． そのような理由で，本書では AWS コンソールを使った AWS の使い方はあまり触れない． AWS のドキュメンテーションには，たくさんの チュートリアル が用意されており，コンソール画面から様々な操作を行う方法が記述されているので，興味がある読者はそちらを参照されたい．
API からリソースを操作する ​
API (Application Programming Interface) を使うことで，コマンドを AWS に送信し，クラウドのリソースの操作をすることができる． API とは，端的に言えば AWS が公開しているコマンドの一覧であり，GET, POST, DELETE などの REST API から構成されている (REST API については REST API で簡単に解説する)． が，直接 REST API を入力するのは面倒であるので，その手間を解消するための様々なツールが提供されている．
例えば， AWS CLI は， UNIX コンソールから AWS API を実行するための CLI (Command Line Interface) である． CLI に加えて，いろいろなプログラミング言語での SDK (Software Development Kit) が提供されている．以下に一例を挙げる．
具体的な API の使用例を見てみよう．
S3 に新しい保存領域 (Bucket (バケット) とよばれる) を追加したいとしよう． AWS CLI を使った場合は，次のようなコマンドを打てばよい．
shell
$ aws s3 mb s3://my-bucket --region ap-northeast-1
$ aws s3 mb s3://my-bucket --region ap-northeast-1
上記のコマンドは， my-bucket という名前のバケットを， ap-northeast-1 のリージョンに作成する．
Python からこれと同じ操作を実行するには， boto3 ライブラリを使って，次のようなスクリプトを実行する．
python
import boto3
+import{_ as n}from"./chunks/earth_from_earth.8c108f53.js";import{_ as l,a as p,b as o,c as e}from"./chunks/terminal.94539704.js";import{_ as t,a as r,b as c,c as y,d as i,e as F,f as A,g as B,h as D,i as C,j as d}from"./chunks/iac.2263bc12.js";import{_ as s,a}from"./chunks/VPC.e1acca4d.js";import{_ as u,a as m,b,c as h,d as g,e as f,f as E}from"./chunks/ec2_keypair_console.fc89ef69.js";import{_,a as v}from"./chunks/cnn.a8836fd9.js";import{_ as k,a as q,b as S,c as w,d as P,e as I,f as x,g as T,h as W,i as $,j as L,k as G,l as N,m as U,n as R,o as X,p as M}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as H,a as z,b as O,c as j,d as V,e as K,f as Y,g as J}from"./chunks/ecs.73d77e6a.js";import{_ as Q,a as Z,b as ss,c as as,d as ns,e as ls,f as ps,g as os,h as es,i as ts,j as rs}from"./chunks/ask_many_output.0bb19110.js";import{_ as cs,a as ys,b as is,c as Fs,d as As,e as Bs,f as Ds,g as Cs,h as ds,i as us,j as ms,k as bs,l as hs,m as gs,n as fs,o as Es,p as _s,q as vs,r as ks,s as qs}from"./chunks/cloud_development.e43e5d4a.js";import{_ as Ss,a as ws}from"./chunks/rest_api.602a6e96.js";import{_ as Ps,a as Is,b as xs,c as Ts,d as Ws}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as $s,a as Ls,b as Gs,c as Ns,d as Us,e as Rs,f as Xs,g as Ms,h as Hs,i as zs,j as Os,k as js,l as Vs}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as Ks,a as Ys,b as Js,c as Qs,d as Zs,e as sa,f as aa,g as na,h as la}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as pa,a as oa,b as ea,c as ta,d as ra,e as ca,f as ya,g as ia,h as Fa,i as Aa,j as Ba,k as Da,l as Ca,m as da,n as ua,o as ma}from"./chunks/venv_shell.dc2530bc.js";import{_ as ba,c as ha,o as ga,R as fa}from"./chunks/framework.a1bac067.js";const Ya=JSON.parse('{"title":"はじめに!","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/main.md","filePath":"development/aws/main.md","lastUpdated":1684859551000}'),Ea={name:"development/aws/main.md"},_a=fa('
はじめに! ​
本書の目的・内容 ​
本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．
本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．
とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．
本書は以下のような三部構成になっている．
本書の構成
テーマハンズオン
第一部 (1章-4章)
クラウドの基礎
  • AWSに自分のサーバーを立ち上げる
第二部 (5章-9章)
クラウドを活用した機械学習
  • AWS と Jupyter を使って始めるディープラーニング
  • スケーラブルな自動質問回答ボットを作る
  • 並列化されたハイパーパラメータサーチの実装
第三部 (10章-13章)
サーバーレスアーキテクチャ入門
  • Lambda, DynamoDB, S3 の演習
  • 俳句を投稿する SNS "Bashoutter" を作る
第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．
第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．
第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．
これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．
本書のフィロソフィー ​
本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．
どういうことか？
ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．
ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．
そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．
決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．
宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)
AWS アカウント ​
本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (AWS アカウントの取得) に簡単に記載したので，必要に応じて参照していただきたい．
AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．
また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．
環境構築 ​
本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (Appendix: 環境構築) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．
  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (WSL のインストール 参照)．
  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については Docker のインストール を参照のこと．
  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は Python クイックガイド 参照のこと．
  • Node.js: version 12.0 以上 をインストールする．
  • AWS CLI: Version 2 をインストールする． インストール手順については AWS CLI のインストール 参照のこと．
  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については AWS CDK のインストール 参照のこと．
  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については AWS CLI のインストール 参照のこと．
ハンズオン実行用の Docker Image ​
Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．
次のコマンドで起動する．
shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc
この Docker image の使い方や詳細は ハンズオン実行用の Docker image の使い方 に記載している．
前提知識 ​
本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．
  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．
  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．
講義に関連する資料 ​
ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．
https://github.com/tomomano/learn-aws-by-coding
本書で使用するノーテーションなど ​
  • コードやシェルのコマンドは monospace letter で記述する．
  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．
また，以下のような形式で注意やチップスを提供する．
追加のコメントなどを記す．
発展的な議論やアイディアなどを紹介する．
陥りやすいミスなどの注意事項を述べる．
絶対に犯してはならないミスを指摘する．
クラウド概論 ​
クラウドとは？ ​
Cloud
クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．
学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．
The NIST Definition of Cloud Computing
これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．
  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．
  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．
  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．
  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．
  • Measured service 計算資源の利用量を計測・監視することができる．
…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．
個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．
クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．
このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．
著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．
クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．
先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．
  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．
  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．
  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．
本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．
SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (Hands-on #6: Bashoutter) などの演習を提供する．
なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については Hands-on #5: サーバーレス入門 などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．
最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．
一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．
第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．
本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．
なぜクラウドを使うのか？ ​
上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．
  1. 自由にサーバーのサイズをスケールできる
    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない
    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0
    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．
    クラウドと自前計算機環境の経済的コストの比較
とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．
ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．
  1. クラウドは賢く使わないといけない
    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ
    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ
    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．
Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？
Terminal
この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．
時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．
最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．
AWS 入門 ​
AWS とは？ ​
本書では，クラウドの実践を行うプラットフォームとして， AWS を用いる． 実践にあたって，最低限必要な AWS の知識を本章では解説しよう．
AWS (Amazon Web Services) は Amazon 社が提供する総合的なクラウドプラットフォームである． AWS は Amazon 社が持つ膨大な計算リソースを貸し出すクラウドサービスとして，2006 年に誕生した． 2021 年では，クラウドプロバイダーとして最大のマーケットシェア (約 32%) を保持している (参照)． Netflix や Slack をはじめとした多くのウェブ関連のサービスで，一部または全てのサーバーリソースが AWS から提供されているとのことである． よって，知らないうちに AWS の恩恵にあずかっている人も少なくないはずだ．
最大のシェアをもつだけに，機能・サービスの幅広さはほかのクラウドプラットフォームと比べ抜きんでている． また，利用者数が多いことを反映して，公式あるいはサードパーティによる技術紹介記事が数多くウェブ上に存在しているだけでなく，ライブラリのユーザーコミュニティも大きく問題解決が捗るのも魅力の一つだ． 初期のころウェブビジネスを行う企業がユーザーの大半を占めていたが，最近は大学などでの科学研究用途としても頻繁に用いられるようになってきている．
AWS の機能・サービス ​
figure_title は，執筆時点において AWS で提供されている主要な機能・サービスの一覧である．
AWSで提供されている主要なサービス一覧
計算，ストレージ，データベース，ネットワーク，セキュリティなど，クラウドの構築に必要な様々な要素が独立したコンポーネントとして提供されている． 基本的に，これらを組み合わせることで一つのクラウドシステムができあがる．
また，機械学習・音声認識・AR/VR など，特定のアプリケーションにパッケージ済みのサービスも提供されている． これらを合計すると全部で 170 個以上のサービスが提供されているとのことである (参照)．
AWS の初心者が陥りがちなのは，大量のサービスの数に圧倒され，どこから手をつけたらよいのかわからなくなる，という状況である． たくさんのサービスの中から，どのサービスをどの順番で学んでいったらいいのか，その道筋すら明らかでなく，大きな参入障壁となっていることは間違いない． だが実のところ， AWS の基本的な構成要素はそのうちの数個のみに限られる． 基本要素となる機能の使い方を知れば， AWS のおおよそのリソースを使いこなすことが可能になる． ほかの機能の多くは，基本の要素を組み合わせて特定のアプリケーションに特化したパッケージとして AWS が用意したものである． そのポイントを認知することが， AWS の学習の最初のステップである．
ここでは， AWS 上でクラウドシステムを構築するときの基本となる構成要素を列挙する． これらは後のハンズオンで実際にプログラムを書きながら体験する． 現時点では，名前だけでも頭の片隅に記憶してもらえればよい．
計算 ​
S3 EC2 (Elastic Compute Cloud) 様々なスペックの仮想マシンを作成し，計算を実行することができる． クラウドの最も基本となる構成要素である． Hands-on #1: 初めての EC2 インスタンスを起動する, Hands-on #2: AWS でディープラーニングを実践, Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する で詳しく触れる．
S3 Lambda Function as a Service (FaaS) とよばれる，小さな計算をサーバーなしで実行するためのサービス． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
ストレージ ​
S3 EBS (Elastic Block Store) EC2 に付与することのできる仮想データドライブ． いわゆる"普通の"(一般的な OS で使われている)ファイルシステムを思い浮かべてくれたらよい．
S3 S3 (Simple Storage Service) Object Storage とよばれる，API を使ってデータの読み書きを行う，いうなれば”クラウド・ネイティブ”なデータの格納システムである． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
データベース ​
S3 DynamoDB NoSQL 型のデータベースサービス (知っている人は mongoDB などを思い浮かべたらよい)． サーバーレスアーキテクチャの章 (Serverless architecture) で詳しく解説する．
ネットワーク ​
S3 VPC(Virtual Private Cloud) AWS 上に仮想ネットワーク環境を作成し，仮想サーバー間の接続を定義したり，外部からのアクセスなどを管理する． EC2 は VPC の内部に配置されなければならない．
API Gateway S3 API のエンドポイントとバックエンドのサービス (Lambda など) を接続する際に用いる，リバースプロキシとしての役割を担う． Hands-on #6: Bashoutter で詳しく解説する．
Region と Availability Zone ​
AWS を使用する際に知っておかなければならない重要な概念として， リージョン (Region) と Availability Zone (AZ) がある (figure_title)． 以下ではこの概念について簡単に記述する．
AWSにおける Region と Availability Zones
リージョン (Region) とは，おおまかに言うとデータセンターの所在地のことである． 執筆時点において， AWS は世界の 25 の国と地域でデータセンターを所有している． figure_title は執筆時点で利用できるリージョンの世界地図を示している． 日本では東京と大阪にデータセンターがある． 各リージョンには固有の ID がついており，例えば東京は ap-northeast-1, 米国オハイオ州は us-east-2，などと定義されている．
Regions in AWS(出典: https://aws.amazon.com/about-aws/global-infrastructure/)
AWS コンソールにログインすると，画面右上のメニューバーでリージョンを選択することができる(figure_title, 赤丸で囲った箇所)． EC2, S3 などの AWS のリソースは，リージョンごとに完全に独立である． したがって，リソースを新たにデプロイする際，あるいはデプロイ済みのリソースを閲覧する際は，コンソールのリージョンが正しく設定されているか，確認する必要がある． ウェブビジネスを展開する場合などは，世界の各地にクラウドを展開する必要があるが，個人的な研究用途として用いる場合は，最寄りのリージョン (i.e. 東京) を使えば基本的に問題ない．
AWSコンソールでリージョンを選択
Avaialibity Zone (AZ) とは，リージョン内で地理的に隔離されたデータセンターのことである． それぞれのリージョンは 2 個以上の AZ を有しており，もし一つの AZ で火災や停電などが起きた場合でも，ほかの AZ がその障害をカバーすることができる． また， AZ 間は高速な AWS 専用ネットワーク回線で結ばれているため， AZ 間のデータ転送は極めて早い． AZ は，ビジネスなどでサーバーダウンが許容されない場合などに注意すべき概念であり，個人的な用途で使う限りにおいてはあまり深く考慮する必要はない．言葉の意味だけ知っておけば十分である．
AWS を使用する際，どこのリージョンを指定するのがよいのだろうか？ インターネットの接続速度の観点からは，地理的に一番近いリージョンを使用するのが一般的によいだろう． 一方， EC2 の利用料などはリージョンごとに価格設定が若干 (10-20%程度) 異なる． したがって，自分が最も頻繁に利用するサービスの価格が最も安く設定されているリージョンを選択する，というのも重要な視点である． また，いくつかのサービスは，特定のリージョンで利用できない場合もある． これらのポイントから総合的に判断して使用するリージョンを決めると良い．
AWS Educate を利用している読者へ
執筆時点において，AWS Educate による Starter Account を使用している場合は us-east-1 region のみ利用できる (参照)．
AWS でのクラウド開発 ​
AWS のクラウドの全体像がわかってきたところで，次のトピックとして，どのようにして AWS 上にクラウドの開発を行い，展開していくかについての概略を解説しよう．
AWS のリソースを追加・編集・削除するなどの操作を実行するには，コンソールを用いる方法と，API を用いる方法の，二つの経路がある．
コンソール画面からリソースを操作する ​
AWS のアカウントにログインすると，まず最初に表示されるのがAWS コンソールである (figure_title)．
AWSマネージメントコンソール画面
コンソールを使うことで， EC2 のインスタンスを立ち上げたり，S3 のデータを追加・削除したり，ログを閲覧したりなど，AWS 上のあらゆるリソースの操作を GUI (Graphical User Interface) を通して実行することができる． 初めて触る機能をポチポチと試したり，デバッグを行うときなどにとても便利である．
コンソールはさらっと機能を試したり，開発中のクラウドのデバッグをするときには便利なのであるが，実際にクラウドの開発をする場面でこれを直接いじることはあまりない． むしろ，次に紹介する API を使用して，プログラムとしてクラウドのリソースを記述することで開発を行うのが一般的である． そのような理由で，本書では AWS コンソールを使った AWS の使い方はあまり触れない． AWS のドキュメンテーションには，たくさんの チュートリアル が用意されており，コンソール画面から様々な操作を行う方法が記述されているので，興味がある読者はそちらを参照されたい．
API からリソースを操作する ​
API (Application Programming Interface) を使うことで，コマンドを AWS に送信し，クラウドのリソースの操作をすることができる． API とは，端的に言えば AWS が公開しているコマンドの一覧であり，GET, POST, DELETE などの REST API から構成されている (REST API については REST API で簡単に解説する)． が，直接 REST API を入力するのは面倒であるので，その手間を解消するための様々なツールが提供されている．
例えば， AWS CLI は， UNIX コンソールから AWS API を実行するための CLI (Command Line Interface) である． CLI に加えて，いろいろなプログラミング言語での SDK (Software Development Kit) が提供されている．以下に一例を挙げる．
具体的な API の使用例を見てみよう．
S3 に新しい保存領域 (Bucket (バケット) とよばれる) を追加したいとしよう． AWS CLI を使った場合は，次のようなコマンドを打てばよい．
shell
$ aws s3 mb s3://my-bucket --region ap-northeast-1
$ aws s3 mb s3://my-bucket --region ap-northeast-1
上記のコマンドは， my-bucket という名前のバケットを， ap-northeast-1 のリージョンに作成する．
Python からこれと同じ操作を実行するには， boto3 ライブラリを使って，次のようなスクリプトを実行する．
python
import boto3
 
 s3_client = boto3.client("s3", region_name="ap-northeast-1")
 s3_client.create_bucket(Bucket="my-bucket")
import boto3
diff --git a/assets/development_aws_main.md.dd2f6ba8.lean.js b/assets/development_aws_main.md.cc9bca88.lean.js
similarity index 96%
rename from assets/development_aws_main.md.dd2f6ba8.lean.js
rename to assets/development_aws_main.md.cc9bca88.lean.js
index d4c88bba..04013fa9 100644
--- a/assets/development_aws_main.md.dd2f6ba8.lean.js
+++ b/assets/development_aws_main.md.cc9bca88.lean.js
@@ -1 +1 @@
-import{_ as n}from"./chunks/earth_from_earth.8c108f53.js";import{_ as l,a as p,b as o,c as e}from"./chunks/terminal.94539704.js";import{_ as t,a as r,b as c,c as y,d as i,e as F,f as A,g as B,h as D,i as C,j as d}from"./chunks/iac.2263bc12.js";import{_ as s,a}from"./chunks/VPC.e1acca4d.js";import{_ as u,a as m,b,c as h,d as g,e as f,f as E}from"./chunks/ec2_keypair_console.fc89ef69.js";import{_,a as v}from"./chunks/cnn.a8836fd9.js";import{_ as k,a as q,b as S,c as w,d as P,e as I,f as x,g as T,h as W,i as $,j as L,k as G,l as N,m as U,n as R,o as X,p as M}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as H,a as z,b as O,c as j,d as V,e as K,f as Y,g as J}from"./chunks/ecs.73d77e6a.js";import{_ as Q,a as Z,b as ss,c as as,d as ns,e as ls,f as ps,g as os,h as es,i as ts,j as rs}from"./chunks/ask_many_output.0bb19110.js";import{_ as cs,a as ys,b as is,c as Fs,d as As,e as Bs,f as Ds,g as Cs,h as ds,i as us,j as ms,k as bs,l as hs,m as gs,n as fs,o as Es,p as _s,q as vs,r as ks,s as qs}from"./chunks/cloud_development.e43e5d4a.js";import{_ as Ss,a as ws}from"./chunks/rest_api.602a6e96.js";import{_ as Ps,a as Is,b as xs,c as Ts,d as Ws}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as $s,a as Ls,b as Gs,c as Ns,d as Us,e as Rs,f as Xs,g as Ms,h as Hs,i as zs,j as Os,k as js,l as Vs}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as Ks,a as Ys,b as Js,c as Qs,d as Zs,e as sa,f as aa,g as na,h as la}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as pa,a as oa,b as ea,c as ta,d as ra,e as ca,f as ya,g as ia,h as Fa,i as Aa,j as Ba,k as Da,l as Ca,m as da,n as ua,o as ma}from"./chunks/venv_shell.dc2530bc.js";import{_ as ba,c as ha,o as ga,R as fa}from"./chunks/framework.a1bac067.js";const Ya=JSON.parse('{"title":"はじめに!","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/main.md","filePath":"development/aws/main.md","lastUpdated":1684858945000}'),Ea={name:"development/aws/main.md"},_a=fa("",1319),va=[_a];function ka(qa,Sa,wa,Pa,Ia,xa){return ga(),ha("div",null,va)}const Ja=ba(Ea,[["render",ka]]);export{Ya as __pageData,Ja as default};
+import{_ as n}from"./chunks/earth_from_earth.8c108f53.js";import{_ as l,a as p,b as o,c as e}from"./chunks/terminal.94539704.js";import{_ as t,a as r,b as c,c as y,d as i,e as F,f as A,g as B,h as D,i as C,j as d}from"./chunks/iac.2263bc12.js";import{_ as s,a}from"./chunks/VPC.e1acca4d.js";import{_ as u,a as m,b,c as h,d as g,e as f,f as E}from"./chunks/ec2_keypair_console.fc89ef69.js";import{_,a as v}from"./chunks/cnn.a8836fd9.js";import{_ as k,a as q,b as S,c as w,d as P,e as I,f as x,g as T,h as W,i as $,j as L,k as G,l as N,m as U,n as R,o as X,p as M}from"./chunks/mnist_prediction.4ba5b405.js";import{_ as H,a as z,b as O,c as j,d as V,e as K,f as Y,g as J}from"./chunks/ecs.73d77e6a.js";import{_ as Q,a as Z,b as ss,c as as,d as ns,e as ls,f as ps,g as os,h as es,i as ts,j as rs}from"./chunks/ask_many_output.0bb19110.js";import{_ as cs,a as ys,b as is,c as Fs,d as As,e as Bs,f as Ds,g as Cs,h as ds,i as us,j as ms,k as bs,l as hs,m as gs,n as fs,o as Es,p as _s,q as vs,r as ks,s as qs}from"./chunks/cloud_development.e43e5d4a.js";import{_ as Ss,a as ws}from"./chunks/rest_api.602a6e96.js";import{_ as Ps,a as Is,b as xs,c as Ts,d as Ws}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as $s,a as Ls,b as Gs,c as Ns,d as Us,e as Rs,f as Xs,g as Ms,h as Hs,i as zs,j as Os,k as js,l as Vs}from"./chunks/s3_bucket_filelist.89f3f384.js";import{_ as Ks,a as Ys,b as Js,c as Qs,d as Zs,e as sa,f as aa,g as na,h as la}from"./chunks/bashoutter_2.cd8a2a95.js";import{_ as pa,a as oa,b as ea,c as ta,d as ra,e as ca,f as ya,g as ia,h as Fa,i as Aa,j as Ba,k as Da,l as Ca,m as da,n as ua,o as ma}from"./chunks/venv_shell.dc2530bc.js";import{_ as ba,c as ha,o as ga,R as fa}from"./chunks/framework.a1bac067.js";const Ya=JSON.parse('{"title":"はじめに!","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/main.md","filePath":"development/aws/main.md","lastUpdated":1684859551000}'),Ea={name:"development/aws/main.md"},_a=fa("",1319),va=[_a];function ka(qa,Sa,wa,Pa,Ia,xa){return ga(),ha("div",null,va)}const Ja=ba(Ea,[["render",ka]]);export{Ya as __pageData,Ja as default};
diff --git a/assets/development_aws_scientific-computing.md.6e6683eb.js b/assets/development_aws_scientific-computing.md.44b26ae4.js
similarity index 99%
rename from assets/development_aws_scientific-computing.md.6e6683eb.js
rename to assets/development_aws_scientific-computing.md.44b26ae4.js
index 62dad52f..5bbad4c2 100644
--- a/assets/development_aws_scientific-computing.md.6e6683eb.js
+++ b/assets/development_aws_scientific-computing.md.44b26ae4.js
@@ -1 +1 @@
-import{_ as t,a as e}from"./chunks/cnn.a8836fd9.js";import{_ as a,c as p,o as r,R as o}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"クラウドで行う科学計算・機械学習","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/scientific-computing.md","filePath":"development/aws/scientific-computing.md","lastUpdated":1684858945000}'),d={name:"development/aws/scientific-computing.md"},n=o('
クラウドで行う科学計算・機械学習 ​
計算機が発達した現代では，計算機によるシミュレーションやビッグデータの解析は，科学・エンジニアリングの研究の主要な柱である． これらの大規模な計算を実行するには，クラウドは最適である． 本章から始まる第二部では，どのようにしてクラウド上で科学計算を実行するのかを，ハンズオンとともに体験してもらう． 科学計算の具体的な題材として，今回は機械学習(深層学習)を取り上げる．
なお，本書では PyTorch ライブラリを使って深層学習のアルゴリズムを実装するが，深層学習および PyTorch の知識は不要である． 講義ではなぜ・どうやって深層学習をクラウドで実行するかに主眼を置いているので，実行するプログラムの詳細には立ち入らない． 将来，自分で深層学習を使う機会が来たときに，詳しく学んでもらいたい．
なぜ機械学習をクラウドで行うのか？ ​
2010 年頃に始まった第三次 AI ブームのおかげで，学術研究だけでなく社会・ビジネスの文脈でも機械学習に高い関心が寄せられている． とくに，深層学習 (ディープラーニング) とよばれる多層のレイヤーからなるニューラルネットワークを用いたアルゴリズムは，画像認識や自然言語処理などの分野で圧倒的に高い性能を実現し，革命をもたらしている．
深層学習の特徴は，なんといってもそのパラメータの多さである． 層が深くなるほど，層間のニューロンを結ぶ重みパラメータの数が増大していく． たとえば，最新の言語モデルである GPT-3 には1750 億個ものパラメータが含まれている． このような膨大なパラメータを有することで，深層学習は高い表現力と汎化性能を実現しているのである．
GPT-3 に限らず，最近の SOTA (State-of-the-art) の性能を達成するニューラルネットワークでは，百万から億のオーダーのパラメータを内包することは頻繁になってきている． そのような巨大なニューラルネットを訓練 (最適化) させるのは，当然のことながら膨大な計算コストがかかる． 結果として，ひとつの計算機では丸一日以上の時間がかかる場合も珍しくない． 深層学習の発展の速度は目覚ましく，研究・ビジネス両方の観点からも，いかにスループットよくニューラルネットワークの最適化を行えるかが鍵となってくる． そのような問題を解決するのにとても有効な手段が，クラウドである！ ??? でその片鱗を見たように，クラウドを使用することでゼロから数千に至るまでの数のインスタンスを動的に起動し，並列に計算を実行することができる． さらに，深層学習を加速させる目的で，深層学習の演算に専用設計された計算チップ (GPU など) がある． クラウドを利用すると，そのような専用計算チップも無尽蔵に利用することができる． 事実，先述した GPT-3 の学習も，詳細は明かされていないが，Microsoft 社のクラウドを使って行われたと報告されている．
GPU による深層学習の高速化 ​
深層学習の計算で欠かすことのできない技術として， GPU (Graphics Processing Unit) について少し説明する．
GPU は，その名のとおり，元々はコンピュータグラフィックスを出力するための専用計算チップである． CPU (Central Processing Unit) に対し，グラフィックスの演算に特化した設計がなされている． 身近なところでは， XBox や PS5 などのゲームコンソールなどに搭載されているし，ハイエンドなノート型・デスクトップ型計算機にも搭載されていることがある． コンピュータグラフィックスでは，スクリーンにアレイ状に並んだ数百万個の画素をビデオレート (30 fps) 以上で処理する必要がある． そのため，GPU はコアあたりの演算能力は比較的小さいかわりに，チップあたり数百から数千のコアを搭載しており (figure_title)，スクリーンの画素を並列的に処理することで，リアルタイムでの描画を実現している．
GPUのアーキテクチャ．GPUには数百から数千の独立した計算コアが搭載されている． (画像出典: https://devblogs.nvidia.com/nvidia-turing-architecture-in-depth/)
このように，コンピュータグラフィクスの目的で生まれた GPU だが，2010 年前後から，その高い並列計算能力をグラフィックス以外の計算 (科学計算など) に用いるという流れ (General-purpose computing on GPU; GPGPU) が生まれた． GPU のコアは，その設計から，行列の計算など，単純かつ規則的な演算が得意であり，そのような演算に対しては数個程度のコアしかもたない CPU に比べて圧倒的に高い計算速度を実現することができる． 現在では GPGPU は分子動力学や気象シミュレーション，そして機械学習など多くの分野で使われている．
ディープラーニングで最も頻繁に起こる演算が，ニューロンの出力を次の層のニューロンに伝える畳み込み (Convolution) 演算である (figure_title)． 畳み込み演算は，まさに GPU が得意とする演算であり， CPU ではなく GPU を用いることで学習を飛躍的に (最大で数百倍程度) 加速させることができる．
ニューラルネットワークにおける畳み込み演算．
このように GPU は機械学習の計算で欠かせないものであるが，なかなか高価である． たとえば，科学計算・機械学習に専用設計された NVIDIA 社の Tesla V100 というチップは，一台で約百万円の価格が設定されている． 機械学習を始めるのに，いきなり百万円の投資はなかなか大きい． だが，クラウドを使えば，初期コスト０で GPU を使用することができる．
機械学習を行うのに， V100 が必ずしも必要というわけではない． むしろ，研究者などでしばしば行われるのは，コンピュータゲームに使われるグラフィックス用の GPU を買ってきて (NVIDIA GeForce シリーズなど)，開発のときはをそれを用いる，というアプローチである． グラフィックス用のいわゆる"コンシューマ GPU"は，市場の需要が大きいおかげで，10 万円前後の価格で購入することができる． V100 と比べると，コンシューマ GPU はコアの数が少なかったり，メモリーが小さかったりなどで劣る点があるが， それらを除いては計算能力にとくに制限があるわけではなく，開発の段階では十分な性能である場合がほとんどである． プログラムができあがって，ビッグデータの解析や，モデルをさらに大きくしたいときなどに，クラウドは有効だろう．
クラウドで GPU を使うには， GPU が搭載された EC2 インスタンスタイプ (P3, P2, G3, G4 など) を選択しなければならない． table_title に，代表的な GPU 搭載のインスタンスタイプを挙げる (執筆時点での情報)．
GPUを搭載したEC2インスタンスタイプ
InstanceGPUsGPU modelGPU Mem (GiB)vCPUMem (GiB)Price per hour ($)
p3.2xlarge
1
NVIDIA V100
16
8
61
3.06
p3n.16xlarge
8
NVIDIA V100
128
64
488
24.48
p2.xlarge
1
NVIDIA K80
12
4
61
0.9
g4dn.xlarge
1
NVIDIA T4
16
4
16
0.526
table_title からわかるとおり， CPU のみのインスタンスと比べると少し高い価格設定になっている． また，古い世代の GPU (V100 に対しての K80) はより安価な価格で提供されている． 1 インスタンスあたりの GPU の搭載数は 1 台から最大で 8 台まで選択することが可能である．
GPU を搭載した一番安いインスタンスタイプは， g4dn.xlarge であり，これには廉価かつ省エネルギー設計の NVIDIA T4 が搭載されている． 後のハンズオンでは，このインスタンスを使用して，ディープラーニングの計算を行ってみる．
table_title の価格は us-east-1 のものである． リージョンによって多少価格設定が異なる．
V100 を一台搭載した p3.2xlarge の利用料金は一時間あたり $3.06 である． V100 が約百万円で売られていることを考えると，約 3000 時間 (= 124 日間)，通算で計算を行った場合に，クラウドを使うよりも V100 を自分で買ったほうがお得になる，という計算になる (実際には，自前で V100 を用意する場合は， V100 だけでなく， CPU やネットワーク機器，電気使用料も必要なので，百万円よりもさらにコストがかかる)．
GPT-3 で使われた計算リソースの詳細は論文でも明かされていないのだが， Lambda 社のブログで興味深い考察が行われている (Lambda 社は機械学習に特化したクラウドサービスを提供している)．
記事によると，1750 億のパラメータを訓練するには，一台の GPU (NVIDIA V100) を用いた場合，342 年の月日と 460 万ドルのクラウド利用料が必要となる，とのことである． GPT-3 のチームは，複数の GPU に処理を分散することで現実的な時間のうちに訓練を完了させたのであろうが，このレベルのモデルになってくるとクラウド技術の限界を攻めないと達成できないことは確かである．
深層学習を詳しく勉強したい人には以下の参考書を推薦したい． 深層学習の基礎的な概念や理論は普遍的であるが，この分野は日進月歩なので，常に最新の情報を取り入れることを忘れずに．
',26),i=[n];function l(s,c,h,P,g,_){return r(),p("div",null,i)}const U=a(d,[["render",l]]);export{u as __pageData,U as default};
+import{_ as t,a as e}from"./chunks/cnn.a8836fd9.js";import{_ as a,c as p,o as r,R as o}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"クラウドで行う科学計算・機械学習","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/scientific-computing.md","filePath":"development/aws/scientific-computing.md","lastUpdated":1684859551000}'),d={name:"development/aws/scientific-computing.md"},n=o('
クラウドで行う科学計算・機械学習 ​
計算機が発達した現代では，計算機によるシミュレーションやビッグデータの解析は，科学・エンジニアリングの研究の主要な柱である． これらの大規模な計算を実行するには，クラウドは最適である． 本章から始まる第二部では，どのようにしてクラウド上で科学計算を実行するのかを，ハンズオンとともに体験してもらう． 科学計算の具体的な題材として，今回は機械学習(深層学習)を取り上げる．
なお，本書では PyTorch ライブラリを使って深層学習のアルゴリズムを実装するが，深層学習および PyTorch の知識は不要である． 講義ではなぜ・どうやって深層学習をクラウドで実行するかに主眼を置いているので，実行するプログラムの詳細には立ち入らない． 将来，自分で深層学習を使う機会が来たときに，詳しく学んでもらいたい．
なぜ機械学習をクラウドで行うのか？ ​
2010 年頃に始まった第三次 AI ブームのおかげで，学術研究だけでなく社会・ビジネスの文脈でも機械学習に高い関心が寄せられている． とくに，深層学習 (ディープラーニング) とよばれる多層のレイヤーからなるニューラルネットワークを用いたアルゴリズムは，画像認識や自然言語処理などの分野で圧倒的に高い性能を実現し，革命をもたらしている．
深層学習の特徴は，なんといってもそのパラメータの多さである． 層が深くなるほど，層間のニューロンを結ぶ重みパラメータの数が増大していく． たとえば，最新の言語モデルである GPT-3 には1750 億個ものパラメータが含まれている． このような膨大なパラメータを有することで，深層学習は高い表現力と汎化性能を実現しているのである．
GPT-3 に限らず，最近の SOTA (State-of-the-art) の性能を達成するニューラルネットワークでは，百万から億のオーダーのパラメータを内包することは頻繁になってきている． そのような巨大なニューラルネットを訓練 (最適化) させるのは，当然のことながら膨大な計算コストがかかる． 結果として，ひとつの計算機では丸一日以上の時間がかかる場合も珍しくない． 深層学習の発展の速度は目覚ましく，研究・ビジネス両方の観点からも，いかにスループットよくニューラルネットワークの最適化を行えるかが鍵となってくる． そのような問題を解決するのにとても有効な手段が，クラウドである！ ??? でその片鱗を見たように，クラウドを使用することでゼロから数千に至るまでの数のインスタンスを動的に起動し，並列に計算を実行することができる． さらに，深層学習を加速させる目的で，深層学習の演算に専用設計された計算チップ (GPU など) がある． クラウドを利用すると，そのような専用計算チップも無尽蔵に利用することができる． 事実，先述した GPT-3 の学習も，詳細は明かされていないが，Microsoft 社のクラウドを使って行われたと報告されている．
GPU による深層学習の高速化 ​
深層学習の計算で欠かすことのできない技術として， GPU (Graphics Processing Unit) について少し説明する．
GPU は，その名のとおり，元々はコンピュータグラフィックスを出力するための専用計算チップである． CPU (Central Processing Unit) に対し，グラフィックスの演算に特化した設計がなされている． 身近なところでは， XBox や PS5 などのゲームコンソールなどに搭載されているし，ハイエンドなノート型・デスクトップ型計算機にも搭載されていることがある． コンピュータグラフィックスでは，スクリーンにアレイ状に並んだ数百万個の画素をビデオレート (30 fps) 以上で処理する必要がある． そのため，GPU はコアあたりの演算能力は比較的小さいかわりに，チップあたり数百から数千のコアを搭載しており (figure_title)，スクリーンの画素を並列的に処理することで，リアルタイムでの描画を実現している．
GPUのアーキテクチャ．GPUには数百から数千の独立した計算コアが搭載されている． (画像出典: https://devblogs.nvidia.com/nvidia-turing-architecture-in-depth/)
このように，コンピュータグラフィクスの目的で生まれた GPU だが，2010 年前後から，その高い並列計算能力をグラフィックス以外の計算 (科学計算など) に用いるという流れ (General-purpose computing on GPU; GPGPU) が生まれた． GPU のコアは，その設計から，行列の計算など，単純かつ規則的な演算が得意であり，そのような演算に対しては数個程度のコアしかもたない CPU に比べて圧倒的に高い計算速度を実現することができる． 現在では GPGPU は分子動力学や気象シミュレーション，そして機械学習など多くの分野で使われている．
ディープラーニングで最も頻繁に起こる演算が，ニューロンの出力を次の層のニューロンに伝える畳み込み (Convolution) 演算である (figure_title)． 畳み込み演算は，まさに GPU が得意とする演算であり， CPU ではなく GPU を用いることで学習を飛躍的に (最大で数百倍程度) 加速させることができる．
ニューラルネットワークにおける畳み込み演算．
このように GPU は機械学習の計算で欠かせないものであるが，なかなか高価である． たとえば，科学計算・機械学習に専用設計された NVIDIA 社の Tesla V100 というチップは，一台で約百万円の価格が設定されている． 機械学習を始めるのに，いきなり百万円の投資はなかなか大きい． だが，クラウドを使えば，初期コスト０で GPU を使用することができる．
機械学習を行うのに， V100 が必ずしも必要というわけではない． むしろ，研究者などでしばしば行われるのは，コンピュータゲームに使われるグラフィックス用の GPU を買ってきて (NVIDIA GeForce シリーズなど)，開発のときはをそれを用いる，というアプローチである． グラフィックス用のいわゆる"コンシューマ GPU"は，市場の需要が大きいおかげで，10 万円前後の価格で購入することができる． V100 と比べると，コンシューマ GPU はコアの数が少なかったり，メモリーが小さかったりなどで劣る点があるが， それらを除いては計算能力にとくに制限があるわけではなく，開発の段階では十分な性能である場合がほとんどである． プログラムができあがって，ビッグデータの解析や，モデルをさらに大きくしたいときなどに，クラウドは有効だろう．
クラウドで GPU を使うには， GPU が搭載された EC2 インスタンスタイプ (P3, P2, G3, G4 など) を選択しなければならない． table_title に，代表的な GPU 搭載のインスタンスタイプを挙げる (執筆時点での情報)．
GPUを搭載したEC2インスタンスタイプ
InstanceGPUsGPU modelGPU Mem (GiB)vCPUMem (GiB)Price per hour ($)
p3.2xlarge
1
NVIDIA V100
16
8
61
3.06
p3n.16xlarge
8
NVIDIA V100
128
64
488
24.48
p2.xlarge
1
NVIDIA K80
12
4
61
0.9
g4dn.xlarge
1
NVIDIA T4
16
4
16
0.526
table_title からわかるとおり， CPU のみのインスタンスと比べると少し高い価格設定になっている． また，古い世代の GPU (V100 に対しての K80) はより安価な価格で提供されている． 1 インスタンスあたりの GPU の搭載数は 1 台から最大で 8 台まで選択することが可能である．
GPU を搭載した一番安いインスタンスタイプは， g4dn.xlarge であり，これには廉価かつ省エネルギー設計の NVIDIA T4 が搭載されている． 後のハンズオンでは，このインスタンスを使用して，ディープラーニングの計算を行ってみる．
table_title の価格は us-east-1 のものである． リージョンによって多少価格設定が異なる．
V100 を一台搭載した p3.2xlarge の利用料金は一時間あたり $3.06 である． V100 が約百万円で売られていることを考えると，約 3000 時間 (= 124 日間)，通算で計算を行った場合に，クラウドを使うよりも V100 を自分で買ったほうがお得になる，という計算になる (実際には，自前で V100 を用意する場合は， V100 だけでなく， CPU やネットワーク機器，電気使用料も必要なので，百万円よりもさらにコストがかかる)．
GPT-3 で使われた計算リソースの詳細は論文でも明かされていないのだが， Lambda 社のブログで興味深い考察が行われている (Lambda 社は機械学習に特化したクラウドサービスを提供している)．
記事によると，1750 億のパラメータを訓練するには，一台の GPU (NVIDIA V100) を用いた場合，342 年の月日と 460 万ドルのクラウド利用料が必要となる，とのことである． GPT-3 のチームは，複数の GPU に処理を分散することで現実的な時間のうちに訓練を完了させたのであろうが，このレベルのモデルになってくるとクラウド技術の限界を攻めないと達成できないことは確かである．
深層学習を詳しく勉強したい人には以下の参考書を推薦したい． 深層学習の基礎的な概念や理論は普遍的であるが，この分野は日進月歩なので，常に最新の情報を取り入れることを忘れずに．
',26),i=[n];function l(s,c,h,P,g,_){return r(),p("div",null,i)}const U=a(d,[["render",l]]);export{u as __pageData,U as default};
diff --git a/assets/development_aws_scientific-computing.md.6e6683eb.lean.js b/assets/development_aws_scientific-computing.md.44b26ae4.lean.js
similarity index 77%
rename from assets/development_aws_scientific-computing.md.6e6683eb.lean.js
rename to assets/development_aws_scientific-computing.md.44b26ae4.lean.js
index af35c052..2836379b 100644
--- a/assets/development_aws_scientific-computing.md.6e6683eb.lean.js
+++ b/assets/development_aws_scientific-computing.md.44b26ae4.lean.js
@@ -1 +1 @@
-import{_ as t,a as e}from"./chunks/cnn.a8836fd9.js";import{_ as a,c as p,o as r,R as o}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"クラウドで行う科学計算・機械学習","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/scientific-computing.md","filePath":"development/aws/scientific-computing.md","lastUpdated":1684858945000}'),d={name:"development/aws/scientific-computing.md"},n=o("",26),i=[n];function l(s,c,h,P,g,_){return r(),p("div",null,i)}const U=a(d,[["render",l]]);export{u as __pageData,U as default};
+import{_ as t,a as e}from"./chunks/cnn.a8836fd9.js";import{_ as a,c as p,o as r,R as o}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"クラウドで行う科学計算・機械学習","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/scientific-computing.md","filePath":"development/aws/scientific-computing.md","lastUpdated":1684859551000}'),d={name:"development/aws/scientific-computing.md"},n=o("",26),i=[n];function l(s,c,h,P,g,_){return r(),p("div",null,i)}const U=a(d,[["render",l]]);export{u as __pageData,U as default};
diff --git a/assets/development_aws_serverless.md.7d293d5a.js b/assets/development_aws_serverless.md.954dd7a1.js
similarity index 99%
rename from assets/development_aws_serverless.md.7d293d5a.js
rename to assets/development_aws_serverless.md.954dd7a1.js
index 493a6c3a..987c2048 100644
--- a/assets/development_aws_serverless.md.7d293d5a.js
+++ b/assets/development_aws_serverless.md.954dd7a1.js
@@ -1 +1 @@
-import{_ as e,a,b as t,c as r,d as o}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as s,c as n,o as p,R as l}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Serverless architecture","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/serverless.md","filePath":"development/aws/serverless.md","lastUpdated":1684858945000}'),d={name:"development/aws/serverless.md"},i=l('
Serverless architecture ​
サーバーレスアーキテクチャ (Serverless architecture) あるいは サーバーレスコンピューティング (Serverless computing) とは，従来とは全く異なるアプローチに基づくクラウドシステムの設計方法である． 歴史的には， AWS が 2014 年に発表した Lamba がサーバーレスアーキテクチャの先駆けとされている． その後， Google や Microsoft などのクラウドプラットフォームも同様の機能の提供を開始している． サーバーレスアーキテクチャの利点は，スケーラブルなクラウドシステムを安価かつ簡易に作成できる点であり，近年いたるところで導入が進んでいる．
Serverless とは，文字どおりの意味としてはサーバーなしで計算をするということになるが，それは一体どういう意味だろうか？ サーバーレスについて説明するためには，まずは従来的な， "serverful" とよばれるようなシステムについて解説しなければならない．
Serverful クラウド (従来型) ​
従来的なクラウドシステムのスケッチを figure_title に示す． クライアントから送信されたリクエストは，最初に API サーバーに送られる． API サーバーでは，リクエストの内容に応じてタスクが実行される． タスクには，API サーバーだけで完結できるものもあるが，多くの場合，データベースの読み書きが必要である． データベースには，データベース専用の独立したサーバーマシンが用いられることが一般的である． また，画像や動画などの容量の大きいデータは，また別のストレージサーバーに保存されることが多い． これらの API サーバー，データベースサーバー，ストレージサーバーはそれぞれ独立したサーバーマシンであり， AWS の言葉では EC2 による仮想インスタンスを想定してもらったらよい．
多くのウェブサービスでは，多数のクライアントからのリクエストを処理するため，複数のサーバーマシンがクラウド内で起動し，負荷を分散するような設計がなされている． クライアントから来たリクエストを計算容量に余裕のあるサーバーに振り分けるような操作を Load balancing とよび，そのような操作を担当するマシンのことを Load balancer という．
計算負荷を分散する目的で多数のインスタンスを起動するのはよいのだが，計算負荷が小さすぎてアイドリング状態にあるようではコストと電力の無駄遣いである． したがって，すべてのサーバーが常に目標とする計算負荷を維持するよう，計算の負荷に応じてクラスター内の仮想サーバーの数を動的に増減させるような仕組みが必要である． そのような仕組みをクラスターのスケーリングとよび，負荷の増大に応答して新しい仮想インスタンスをクラスターに追加する操作を scale-out，負荷の減少に応答してインスタンスをシャットダウンする操作を scale-in とよぶ． クラスターのスケーリングは， API サーバーではもちろんのこと，データベースサーバー・ストレージサーバーでも必要になる． ストレージサーバーでは，例えば頻繁にアクセスされるデータはキャッシュ領域に保存したり，データのコピーを複数作るなどのスケーリングが行われる． データベースサーバーも同様に，頻繁にアクセスされるデータのアクセスがパンクしてしまわないよう，分散的な処理が必要となる． このように，クラウドシステム内すべての箇所で，負荷が均一になるような調整が必要であり，開発者は多くの時間をそのチューニングに費やさなければならない． また，サービスの利用者の数などに応じてスケーリングの設定は常に見直される必要があり，継続的な開発が要求される．
さらに問題を複雑にするのは，API サーバーで処理されるべきタスクが，非一様な点である． 非一様であるとは，たとえばタスク A は 3000 ミリ秒の実行時間と 512MB のメモリーを消費し，別のタスク B は 1000 ミリ秒の実行時間と 128MB のメモリーを消費する，というような状況を指している． 一つのサーバーマシンが計算負荷が異なる複数のタスクを処理する場合，クラスターのスケーリングはより複雑になる． この状況をシンプルにするために，１サーバーで実行するタスクは１種類に限る，という設計も可能であるが，そうするとで生まれる弊害も多い (ほとんど使われないタスクに対してもサーバー一台をまるまる割り当てなければならない = ほとんどアイドリング状態になってしまう，など)．
Serverful なクラウドシステム
Serverless クラウドへ ​
Serverful クラウド (従来型) で議論したように，クラスターのスケーリングはクラウドシステムの経済的効率とシステムの安定性を最大化するために必須の作業である． それを反映して，多くの開発者の時間が投資されてきた．
クラスターのスケーリングはすべての開発者が何度も繰り返し行ってきた作業であり，いくつかの側面をテンプレート化し，共通化することができたならば開発のコストを大幅に削減できるだろう． それを実現するには，根本的なレベルからクラウドシステムの設計を考え直す必要がある． スケーリングを前提として考えることで，もっとシンプルで見通しがよいクラウドシステムの設計の仕組みはないだろうか？ そのような動機が，サーバーレスアーキテクチャが誕生する背後にあった．
従来の serverful なシステムでの最大の問題点は，サーバーをまるまる占有してしまうという点にある． すなわち， EC2 インスタンスを起動したとき，そのインスタンスは起動したユーザーだけが使えるものであり，計算のリソース (CPU や RAM) が独占的に割り当てられた状態になる． 固定した計算資源の割り当てがされてしまっているので，インスタンスの計算負荷が 0%であろうが 100%であろうが，均一の使用料金が起動時間に比例して発生する．
サーバーレスアーキテクチャは，このような 独占的に割り当てられた計算リソースというものを完全に廃止することを出発点とする． サーバーレスアーキテクチャでは，計算のリソースは，クラウドプロバイダーがすべて管理する． クライアントは，仮想インスタンスを一台まるごと借りるのではなく，計算のタスクの需要が生まれる毎に，実行したいプログラム・コマンドをクラウドに提出する． クラウドプロバイダーは，自身のもつ巨大な計算リソースから空きを探し，提出されたプログラムを実行し，実行結果をクライアントに返す． 言い換えると，計算リソースのスケーリングやアロケーションなどはクラウドプロバイダーが一手に引き受け，ユーザーはジョブをサブミットすることに注力する，という枠組みである． これを図示すると， figure_title のようになる．
従来のクラウドと Serverless クラウドの比較
サーバーレスクラウドでは，スケーリングはすべてクラウドプロバイダーが引き受けるので，スケーラビリティーが保証されている． クライアントが同時に大量のタスクを送信した場合でも，クラウドプロバイダー側の独自の仕組みによってすべてのタスクが遅延なく実行される． また，サーバーレスクラウドを利用することで，クラウドのコストは実際に使用した計算の総量 (稼働時間) で決定されることになる． これは，計算の実行総量に関わらずインスタンスの起動時間で料金が決定されていた従来のシステムと比べて大きな違いである．
サーバーレスクラウドは，従来のクラウドとは根本から異なったアプローチなので，コードの書き方やシステムの設計が大きく異なる． サーバーレスクラウドを開発・運用するには，サーバーレス固有の概念や用語に精通している必要がある． 以降では，実際にクラウドを動かしながら，サーバーレスをより具体的に体験していこう．
従来型の(仮想インスタンスをたくさん起動するような)クラウドシステムは，賃貸と似ているかもしれない． 部屋を借りるというのは，その部屋でどれだけの時間を過ごそうが，月々の家賃は一定である． 同様に，仮想サーバーも，それがどれほどの計算を行っているかに関わらず，一定の料金が時間ごとに発生する．
一方で，サーバーレスクラウドは，電気・水道・ガス料金 と似ている． こちらは，実際に使用した量に比例して料金が決定されている． サーバーレスクラウドも，実際に計算を行った総時間で料金が決まる仕組みになっている．
サーバーレスクラウドを構成するコンポーネント ​
サーバーレスアーキテクチャの概要がわかってきたところで，ここでは AWS においてサーバーレスクラウドを構成する様々なコンポーネントを紹介していこう． 特に， Lambda, S3, DynamoDB を取り上げ，解説する (figure_title)． サーバーレスクラウドは，これらのコンポーネントを統合することで一つのシステムが出来上がる． ここでは， Lambda，S3，DynamoDB を利用する際に押さえておかなければならない知識を一通り説明しきる都合上，具体的なイメージがわきにくいかもしれない． が，続く ??? でそれぞれについてハンズオン形式で演習を行うので，そこでさらに理解を深めれば大丈夫である．
Lambda, S3, DynamoDB のアイコン
Lambda ​
AWS でサーバーレスコンピューティングの中心を担うのが， Lambda である． Lambda の使い方を figure_title に図示している． Lambda の仕組みはシンプルで，まずユーザーは実行したいプログラムのコードを事前に登録しておく． プログラムは， Python, Node.js, Ruby などの主要な言語がサポートされている． Lambda に登録されたひとつひとつのプログラムを関数 (Function) とよぶ． そして，関数を実行したいときに，invoke コマンドを Lambda に送信する． Lambda では， invoke のリクエストを受け取るとただちに (数ミリセカンドから数百ミリセカンド程度の時間で) プログラムの実行を開始する． そして，実行結果をクライアントやその他の計算機に返す．
AWS Lambda
このように，Lambda では占有された仮想インスタンスは存在せず，実行を待っているプログラムだけがある状態である． invoke のリクエストに応じて，プログラムが AWS の巨大な計算機プールのどこかに配置され，実行される． 同時に複数のリクエストが来た場合でも， AWS はそれらを実行するための計算リソースを割り当て，並列的に処理を行ってくれる． 原理上は，数千から数万のリクエストが同時に来たとしても， Lambda はそれらを同時に実行することができる． このような，占有された仮想サーバーの存在なしに，動的に関数を実行するサービスのことを総称して FaaS (Function as a Service) とよぶ．
Lambda ではそれぞれの関数につき， 128MB から 10240MB のメモリーを使用することができる (執筆時点の仕様)． また，実効的な CPU のパワーはメモリーの量に比例する形で割り当てられる． すなわち，タスクに割り当てたメモリーの量が多ければ多いほど，より多くの CPU リソースが割り当てられることになる (しかし， RAM と CPU パワーの具体的な換算表は AWS からは公開されていない)． 実行時間は 100 ミリ秒の単位で記録され，実行時間に比例して料金が決定される． table_title は Lambda の利用料金表である (執筆時点で ap-north-east1 リージョンを選択した場合)．
Lambda の料金表
Memory (MB)Price per 100ms
128
$0.0000002083
512
$0.0000008333
1024
$0.0000016667
3008
$0.0000048958
実行時間に比例する料金に追加して，リクエストを送信するごとに発生する料金が設定されている． これは，百万回のリクエストにつき $0.2 である． たとえば， 128MB のメモリーを使用する関数を，それぞれ 200 ミリ秒，合計で 100 万回実行した場合， 0.0000002083 * 2 * 10^6 + 0.2 = $0.6 の料金となる． ウェブサーバーのデータベースの更新など簡単な計算であれば，200 ミリ秒程度で実行できる関数も多いことから，100 万回データベースの更新を行ったとしても，たった $0.6 しかコストが発生しないことになる． また，コードが実行されず待機状態になっている場合は，発生する料金は 0 である． このように，実際に意味のある処理が行われた時間にのみ，料金が発生する仕組みになっている．
Lambda は比較的短時間で完了する，反復性の高いタスクの実行に向いている． データベースの読み書きはその典型的な例であるが，そのほかにも，画像のサイズをトリミングしたり，サーバーサイドで定期的に実行されるメンテナンス処理などの利用が考えられる． また，複数の Lambda をリレー式に繋げることも可能で，シンプルな処理を組み合わせることで複雑なロジックを表現することができる．
上述の Lambda の料金計算は，説明のためコストに寄与する要素をいくつか省いている点は承知いただきたい． 例えば， DynamoDB の読み書きに関する料金や，ネットワークの通信にかかわるコストが考慮されていない．
サーバーレスストレージ: S3 ​
サーバーレスの概念は，ストレージにも拡張されている．
従来的なストレージ (ファイルシステム) では，必ずホストとなるマシンと OS が存在しなければならない． したがって，それほどパワーは必要ないまでも，ある程度の CPU リソースを割かなければならない． また，従来的なファイルシステムでは，データ領域のサイズは最初にディスクを初期化するときに決めなければならず，後から容量を増加させることはしばしば困難である (ZFS などのファイルシステムを使えばある程度は自由にファイルシステムのサイズを変更することは可能である)． よって，従来的なクラウドでは，ストレージを借りる際にはあらかじめディスクのサイズを指定せねばならず，ディスクの中身が空であろうと満杯であろうと，同じ利用料金が発生することになる (figure_title)．
Simple Storage Service (S3) は，サーバーレスなストレージシステムを提供する (figure_title)． S3 は従来的なストレージシステムと異なり， OS に"マウントする”という概念はない． 基本的に API を通じてデータの読み書きの操作が行われる． また，データの冗長化や暗号化，バックアップの作成など，通常ならば OS と CPU が介在しなければならない操作も， API を通じて行うことができる． S3 では事前に決められたディスク領域のサイズはなく，データを入れれば入れた分だけ，保存領域は拡大していく (仕様上はペタバイトスケールのデータを保存することが可能である)． ストレージにかかる料金は，保存してあるデータの総容量で決定される．
S3 と従来的なファイルシステムの比較
S3 を利用する際に，料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 説明のため主要な事項のみ取り出している． 詳細は 公式ドキュメンテーション "Amazon S3 pricing" を参照)．
S3 の利用料金
項目料金
Data storage (First 50TB)
$0.023 per GB per month
PUT, COPY, POST, LIST requests (per 1,000 requests)
$0.005
GET, SELECT, and all other requests (per 1,000 requests)
$0.0004
Data Transfer IN To Amazon S3 From Internet
$0
Data Transfer OUT From Amazon S3 To Internet
$0.09 per GB
第一に，データの保存には $0.025 per GB のコストが月ごとに発生する． したがって，1000GB のデータを S3 に一ヵ月保存した場合， $25 の料金が発生することになる． また，PUT, COPY, POST などのリクエスト (=データを書き込む操作) に対しては，データ容量に関係なく，1000 回ごとに $0.005 のコストが発生する． GET, SELECT などのリクエスト (=データを読み込む操作) に対しては，1000 回ごとに $0.0004 のコストが発生する． また， S3 はデータを外に取り出す際の通信にもコストが生じる． 執筆時点では，S3 からインターネットを通じて外部にデータを転送 (data-out) すると $0.09 per GB のコストが発生する． データをインターネットを通じて S3 に入れる (data-in) 通信は無料で行える． また， AWS の 同じ Region 内のサービス (Lambda や EC2 など) にデータを転送するのは無料である． AWS のリージョンをまたいだデータの転送にはコストが発生する． いずれにせよ，サーバーレスの概念に則り，すべての料金が従量課金制で決定される設定になっている．
サーバーレスデータベース: DynamoDB ​
サーバーレスの概念は，データベースにも適用することができる．
ここでいうデータベースとは， Web サービスなどにおけるユーザーや商品の情報を記録しておくための保存領域のことを指している． 従来的に有名なデータベースとしては MySQL, PostgreSQL, MongoDB などが挙げられる． データベースと普通のストレージの違いは，データの検索機能にある． 普通のストレージではデータは単純にディスクに書き込まれるだけだが， データベースでは検索がより効率的になるようなデータの配置がされたり， 頻繁にアクセスされるデータはメモリーにキャッシュされるなどの機能が備わっている． これにより，巨大なデータの中から，興味のある要素を高速に取得することができる．
このような検索機能を実現するには，当然 CPU の存在が必須である． したがって，従来的なデータベースを構築する際は，ストレージ領域に加えて，たくさんの CPU コアを搭載したマシンが用いられることが多い． また，データベースが巨大な場合は複数マシンにまたがった分散型のシステムが設計される． 分散型システムの場合は， Serverful クラウド (従来型) で議論したようにデータベースへのアクセス負荷に応じて適切なスケーリングがなされる必要がある．
DynamoDB は， AWS が提供しているサーバーレスな分散型データベースである． サーバーレスであるので，占有されたデータベース用仮想インスタンスは存在せず， API を通じてデータの書き込み・読み出し・検索などの操作を行う． S3 と同様に，データ保存領域の上限は定められておらず，データを入れれば入れた分だけ，保存領域は拡大していく． また，データベースへの負荷が増減したときのスケーリングは， DynamoDB が自動で行うので，ユーザーは心配する必要はない．
DynamoDB での利用料金の計算はやや複雑なのだが， "On-demand Capacity" というモードで使用した場合の料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 詳細は 公式ドキュメンテーション "Pricing for On-Demand Capacity" を参照)．
DynamoDB の利用料金
項目料金
Write request units
$1.25 per million write request units
Read request units
$0.25 per million read request units
Data storage
$0.25 per GB-month
DynamoDB ではデータの書き込み操作の単位を write request unit とよび，データの読み込み操作の単位を read request unit とよぶ． 基本的に， 1kB 以下のデータを一度書き込むと 1 write request unit を消費し，4kB 以下のデータを一度読み込むと 1 read request unit を消費する (詳しくは 公式ドキュメンテーション "Read/Write Capacity Mode" を参照のこと)． write request units は 100 万回ごとに $1.25, read request units は 100 万回ごとに $0.25 のコストが設定されている． また，保存されたデータ容量に対して $0.25 per GB のコストが月ごとに発生する． DynamoDB は高速な検索機能などを備えたデータベースであるので， GB あたりのストレージコストは S3 に比べ 10 倍程度高い． DynamoDB のデータの転送に関わるコストは，同じリージョン内ならば data-in，data-out ともに $0 である． リージョンをまたいだ通信には別途コストが発生する．
その他のサーバーレスクラウドの構成要素 ​
以上で紹介した Lambda, S3, DynamoDB がサーバーレスクラウドの中で最も使用する頻度が高いサービスになる． その他のサーバーレスクラウドの構成要素を以下に列挙する． いくつかについては，今後のハンズオンを行う中で改めて解説を行う．
  • API Gateway: API を構築する際のルーティングを担う． ??? で取り上げる．
  • Fargate: ??? で触れた Fargate も，サーバーレスクラウドの要素の一部である． Lambda との違いは，Lambda よりも大容量のメモリーや CPU を要するような計算などを行うことができる点が挙げられる．
  • Simple Notification Service (SNS): サーバーレスのサービス間でイベントをやり取りするためのサービス．
  • Step Functions: サーバーレスのサービス間のオーケストレーションを担う．
サーバーレスアーキテクチャは万能か？
この問いへの答えは，筆者は NO であると考える．
ここまで，サーバーレスの利点を強調して説明をしてきたが，まだまだ新しい技術なだけに，欠点，あるいはサーバーフルなシステムに劣る点は数多くある．
大きな欠点を一つあげるとすれば，サーバーレスのシステムは各クラウドプラットフォームに固有なものなので，特定のプラットフォームでしか運用できないシステムになってしまう点であろう． AWS で作成したサーバーレスのシステムを， Google のクラウドに移植するには，かなり大掛かりなプログラムの書き換えが必要になる． 一方， serverful なシステムであれば，プラットフォーム間のマイグレーションは比較的簡単に行うことができる． クラウドプロバイダーとしては，自社のシステムへの依存度を強めることで，顧客を離さないようにするという狙いがあるのだろう…
その他，サーバーレスコンピューティングの欠点や今後の課題などは，次の論文で詳しく議論されている． 興味のある読者はぜひ読んでいただきたい．
',56),c=[i];function h(m,g,u,b,f,_){return p(),n("div",null,c)}const y=s(d,[["render",h]]);export{q as __pageData,y as default};
+import{_ as e,a,b as t,c as r,d as o}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as s,c as n,o as p,R as l}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Serverless architecture","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/serverless.md","filePath":"development/aws/serverless.md","lastUpdated":1684859551000}'),d={name:"development/aws/serverless.md"},i=l('
Serverless architecture ​
サーバーレスアーキテクチャ (Serverless architecture) あるいは サーバーレスコンピューティング (Serverless computing) とは，従来とは全く異なるアプローチに基づくクラウドシステムの設計方法である． 歴史的には， AWS が 2014 年に発表した Lamba がサーバーレスアーキテクチャの先駆けとされている． その後， Google や Microsoft などのクラウドプラットフォームも同様の機能の提供を開始している． サーバーレスアーキテクチャの利点は，スケーラブルなクラウドシステムを安価かつ簡易に作成できる点であり，近年いたるところで導入が進んでいる．
Serverless とは，文字どおりの意味としてはサーバーなしで計算をするということになるが，それは一体どういう意味だろうか？ サーバーレスについて説明するためには，まずは従来的な， "serverful" とよばれるようなシステムについて解説しなければならない．
Serverful クラウド (従来型) ​
従来的なクラウドシステムのスケッチを figure_title に示す． クライアントから送信されたリクエストは，最初に API サーバーに送られる． API サーバーでは，リクエストの内容に応じてタスクが実行される． タスクには，API サーバーだけで完結できるものもあるが，多くの場合，データベースの読み書きが必要である． データベースには，データベース専用の独立したサーバーマシンが用いられることが一般的である． また，画像や動画などの容量の大きいデータは，また別のストレージサーバーに保存されることが多い． これらの API サーバー，データベースサーバー，ストレージサーバーはそれぞれ独立したサーバーマシンであり， AWS の言葉では EC2 による仮想インスタンスを想定してもらったらよい．
多くのウェブサービスでは，多数のクライアントからのリクエストを処理するため，複数のサーバーマシンがクラウド内で起動し，負荷を分散するような設計がなされている． クライアントから来たリクエストを計算容量に余裕のあるサーバーに振り分けるような操作を Load balancing とよび，そのような操作を担当するマシンのことを Load balancer という．
計算負荷を分散する目的で多数のインスタンスを起動するのはよいのだが，計算負荷が小さすぎてアイドリング状態にあるようではコストと電力の無駄遣いである． したがって，すべてのサーバーが常に目標とする計算負荷を維持するよう，計算の負荷に応じてクラスター内の仮想サーバーの数を動的に増減させるような仕組みが必要である． そのような仕組みをクラスターのスケーリングとよび，負荷の増大に応答して新しい仮想インスタンスをクラスターに追加する操作を scale-out，負荷の減少に応答してインスタンスをシャットダウンする操作を scale-in とよぶ． クラスターのスケーリングは， API サーバーではもちろんのこと，データベースサーバー・ストレージサーバーでも必要になる． ストレージサーバーでは，例えば頻繁にアクセスされるデータはキャッシュ領域に保存したり，データのコピーを複数作るなどのスケーリングが行われる． データベースサーバーも同様に，頻繁にアクセスされるデータのアクセスがパンクしてしまわないよう，分散的な処理が必要となる． このように，クラウドシステム内すべての箇所で，負荷が均一になるような調整が必要であり，開発者は多くの時間をそのチューニングに費やさなければならない． また，サービスの利用者の数などに応じてスケーリングの設定は常に見直される必要があり，継続的な開発が要求される．
さらに問題を複雑にするのは，API サーバーで処理されるべきタスクが，非一様な点である． 非一様であるとは，たとえばタスク A は 3000 ミリ秒の実行時間と 512MB のメモリーを消費し，別のタスク B は 1000 ミリ秒の実行時間と 128MB のメモリーを消費する，というような状況を指している． 一つのサーバーマシンが計算負荷が異なる複数のタスクを処理する場合，クラスターのスケーリングはより複雑になる． この状況をシンプルにするために，１サーバーで実行するタスクは１種類に限る，という設計も可能であるが，そうするとで生まれる弊害も多い (ほとんど使われないタスクに対してもサーバー一台をまるまる割り当てなければならない = ほとんどアイドリング状態になってしまう，など)．
Serverful なクラウドシステム
Serverless クラウドへ ​
Serverful クラウド (従来型) で議論したように，クラスターのスケーリングはクラウドシステムの経済的効率とシステムの安定性を最大化するために必須の作業である． それを反映して，多くの開発者の時間が投資されてきた．
クラスターのスケーリングはすべての開発者が何度も繰り返し行ってきた作業であり，いくつかの側面をテンプレート化し，共通化することができたならば開発のコストを大幅に削減できるだろう． それを実現するには，根本的なレベルからクラウドシステムの設計を考え直す必要がある． スケーリングを前提として考えることで，もっとシンプルで見通しがよいクラウドシステムの設計の仕組みはないだろうか？ そのような動機が，サーバーレスアーキテクチャが誕生する背後にあった．
従来の serverful なシステムでの最大の問題点は，サーバーをまるまる占有してしまうという点にある． すなわち， EC2 インスタンスを起動したとき，そのインスタンスは起動したユーザーだけが使えるものであり，計算のリソース (CPU や RAM) が独占的に割り当てられた状態になる． 固定した計算資源の割り当てがされてしまっているので，インスタンスの計算負荷が 0%であろうが 100%であろうが，均一の使用料金が起動時間に比例して発生する．
サーバーレスアーキテクチャは，このような 独占的に割り当てられた計算リソースというものを完全に廃止することを出発点とする． サーバーレスアーキテクチャでは，計算のリソースは，クラウドプロバイダーがすべて管理する． クライアントは，仮想インスタンスを一台まるごと借りるのではなく，計算のタスクの需要が生まれる毎に，実行したいプログラム・コマンドをクラウドに提出する． クラウドプロバイダーは，自身のもつ巨大な計算リソースから空きを探し，提出されたプログラムを実行し，実行結果をクライアントに返す． 言い換えると，計算リソースのスケーリングやアロケーションなどはクラウドプロバイダーが一手に引き受け，ユーザーはジョブをサブミットすることに注力する，という枠組みである． これを図示すると， figure_title のようになる．
従来のクラウドと Serverless クラウドの比較
サーバーレスクラウドでは，スケーリングはすべてクラウドプロバイダーが引き受けるので，スケーラビリティーが保証されている． クライアントが同時に大量のタスクを送信した場合でも，クラウドプロバイダー側の独自の仕組みによってすべてのタスクが遅延なく実行される． また，サーバーレスクラウドを利用することで，クラウドのコストは実際に使用した計算の総量 (稼働時間) で決定されることになる． これは，計算の実行総量に関わらずインスタンスの起動時間で料金が決定されていた従来のシステムと比べて大きな違いである．
サーバーレスクラウドは，従来のクラウドとは根本から異なったアプローチなので，コードの書き方やシステムの設計が大きく異なる． サーバーレスクラウドを開発・運用するには，サーバーレス固有の概念や用語に精通している必要がある． 以降では，実際にクラウドを動かしながら，サーバーレスをより具体的に体験していこう．
従来型の(仮想インスタンスをたくさん起動するような)クラウドシステムは，賃貸と似ているかもしれない． 部屋を借りるというのは，その部屋でどれだけの時間を過ごそうが，月々の家賃は一定である． 同様に，仮想サーバーも，それがどれほどの計算を行っているかに関わらず，一定の料金が時間ごとに発生する．
一方で，サーバーレスクラウドは，電気・水道・ガス料金 と似ている． こちらは，実際に使用した量に比例して料金が決定されている． サーバーレスクラウドも，実際に計算を行った総時間で料金が決まる仕組みになっている．
サーバーレスクラウドを構成するコンポーネント ​
サーバーレスアーキテクチャの概要がわかってきたところで，ここでは AWS においてサーバーレスクラウドを構成する様々なコンポーネントを紹介していこう． 特に， Lambda, S3, DynamoDB を取り上げ，解説する (figure_title)． サーバーレスクラウドは，これらのコンポーネントを統合することで一つのシステムが出来上がる． ここでは， Lambda，S3，DynamoDB を利用する際に押さえておかなければならない知識を一通り説明しきる都合上，具体的なイメージがわきにくいかもしれない． が，続く ??? でそれぞれについてハンズオン形式で演習を行うので，そこでさらに理解を深めれば大丈夫である．
Lambda, S3, DynamoDB のアイコン
Lambda ​
AWS でサーバーレスコンピューティングの中心を担うのが， Lambda である． Lambda の使い方を figure_title に図示している． Lambda の仕組みはシンプルで，まずユーザーは実行したいプログラムのコードを事前に登録しておく． プログラムは， Python, Node.js, Ruby などの主要な言語がサポートされている． Lambda に登録されたひとつひとつのプログラムを関数 (Function) とよぶ． そして，関数を実行したいときに，invoke コマンドを Lambda に送信する． Lambda では， invoke のリクエストを受け取るとただちに (数ミリセカンドから数百ミリセカンド程度の時間で) プログラムの実行を開始する． そして，実行結果をクライアントやその他の計算機に返す．
AWS Lambda
このように，Lambda では占有された仮想インスタンスは存在せず，実行を待っているプログラムだけがある状態である． invoke のリクエストに応じて，プログラムが AWS の巨大な計算機プールのどこかに配置され，実行される． 同時に複数のリクエストが来た場合でも， AWS はそれらを実行するための計算リソースを割り当て，並列的に処理を行ってくれる． 原理上は，数千から数万のリクエストが同時に来たとしても， Lambda はそれらを同時に実行することができる． このような，占有された仮想サーバーの存在なしに，動的に関数を実行するサービスのことを総称して FaaS (Function as a Service) とよぶ．
Lambda ではそれぞれの関数につき， 128MB から 10240MB のメモリーを使用することができる (執筆時点の仕様)． また，実効的な CPU のパワーはメモリーの量に比例する形で割り当てられる． すなわち，タスクに割り当てたメモリーの量が多ければ多いほど，より多くの CPU リソースが割り当てられることになる (しかし， RAM と CPU パワーの具体的な換算表は AWS からは公開されていない)． 実行時間は 100 ミリ秒の単位で記録され，実行時間に比例して料金が決定される． table_title は Lambda の利用料金表である (執筆時点で ap-north-east1 リージョンを選択した場合)．
Lambda の料金表
Memory (MB)Price per 100ms
128
$0.0000002083
512
$0.0000008333
1024
$0.0000016667
3008
$0.0000048958
実行時間に比例する料金に追加して，リクエストを送信するごとに発生する料金が設定されている． これは，百万回のリクエストにつき $0.2 である． たとえば， 128MB のメモリーを使用する関数を，それぞれ 200 ミリ秒，合計で 100 万回実行した場合， 0.0000002083 * 2 * 10^6 + 0.2 = $0.6 の料金となる． ウェブサーバーのデータベースの更新など簡単な計算であれば，200 ミリ秒程度で実行できる関数も多いことから，100 万回データベースの更新を行ったとしても，たった $0.6 しかコストが発生しないことになる． また，コードが実行されず待機状態になっている場合は，発生する料金は 0 である． このように，実際に意味のある処理が行われた時間にのみ，料金が発生する仕組みになっている．
Lambda は比較的短時間で完了する，反復性の高いタスクの実行に向いている． データベースの読み書きはその典型的な例であるが，そのほかにも，画像のサイズをトリミングしたり，サーバーサイドで定期的に実行されるメンテナンス処理などの利用が考えられる． また，複数の Lambda をリレー式に繋げることも可能で，シンプルな処理を組み合わせることで複雑なロジックを表現することができる．
上述の Lambda の料金計算は，説明のためコストに寄与する要素をいくつか省いている点は承知いただきたい． 例えば， DynamoDB の読み書きに関する料金や，ネットワークの通信にかかわるコストが考慮されていない．
サーバーレスストレージ: S3 ​
サーバーレスの概念は，ストレージにも拡張されている．
従来的なストレージ (ファイルシステム) では，必ずホストとなるマシンと OS が存在しなければならない． したがって，それほどパワーは必要ないまでも，ある程度の CPU リソースを割かなければならない． また，従来的なファイルシステムでは，データ領域のサイズは最初にディスクを初期化するときに決めなければならず，後から容量を増加させることはしばしば困難である (ZFS などのファイルシステムを使えばある程度は自由にファイルシステムのサイズを変更することは可能である)． よって，従来的なクラウドでは，ストレージを借りる際にはあらかじめディスクのサイズを指定せねばならず，ディスクの中身が空であろうと満杯であろうと，同じ利用料金が発生することになる (figure_title)．
Simple Storage Service (S3) は，サーバーレスなストレージシステムを提供する (figure_title)． S3 は従来的なストレージシステムと異なり， OS に"マウントする”という概念はない． 基本的に API を通じてデータの読み書きの操作が行われる． また，データの冗長化や暗号化，バックアップの作成など，通常ならば OS と CPU が介在しなければならない操作も， API を通じて行うことができる． S3 では事前に決められたディスク領域のサイズはなく，データを入れれば入れた分だけ，保存領域は拡大していく (仕様上はペタバイトスケールのデータを保存することが可能である)． ストレージにかかる料金は，保存してあるデータの総容量で決定される．
S3 と従来的なファイルシステムの比較
S3 を利用する際に，料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 説明のため主要な事項のみ取り出している． 詳細は 公式ドキュメンテーション "Amazon S3 pricing" を参照)．
S3 の利用料金
項目料金
Data storage (First 50TB)
$0.023 per GB per month
PUT, COPY, POST, LIST requests (per 1,000 requests)
$0.005
GET, SELECT, and all other requests (per 1,000 requests)
$0.0004
Data Transfer IN To Amazon S3 From Internet
$0
Data Transfer OUT From Amazon S3 To Internet
$0.09 per GB
第一に，データの保存には $0.025 per GB のコストが月ごとに発生する． したがって，1000GB のデータを S3 に一ヵ月保存した場合， $25 の料金が発生することになる． また，PUT, COPY, POST などのリクエスト (=データを書き込む操作) に対しては，データ容量に関係なく，1000 回ごとに $0.005 のコストが発生する． GET, SELECT などのリクエスト (=データを読み込む操作) に対しては，1000 回ごとに $0.0004 のコストが発生する． また， S3 はデータを外に取り出す際の通信にもコストが生じる． 執筆時点では，S3 からインターネットを通じて外部にデータを転送 (data-out) すると $0.09 per GB のコストが発生する． データをインターネットを通じて S3 に入れる (data-in) 通信は無料で行える． また， AWS の 同じ Region 内のサービス (Lambda や EC2 など) にデータを転送するのは無料である． AWS のリージョンをまたいだデータの転送にはコストが発生する． いずれにせよ，サーバーレスの概念に則り，すべての料金が従量課金制で決定される設定になっている．
サーバーレスデータベース: DynamoDB ​
サーバーレスの概念は，データベースにも適用することができる．
ここでいうデータベースとは， Web サービスなどにおけるユーザーや商品の情報を記録しておくための保存領域のことを指している． 従来的に有名なデータベースとしては MySQL, PostgreSQL, MongoDB などが挙げられる． データベースと普通のストレージの違いは，データの検索機能にある． 普通のストレージではデータは単純にディスクに書き込まれるだけだが， データベースでは検索がより効率的になるようなデータの配置がされたり， 頻繁にアクセスされるデータはメモリーにキャッシュされるなどの機能が備わっている． これにより，巨大なデータの中から，興味のある要素を高速に取得することができる．
このような検索機能を実現するには，当然 CPU の存在が必須である． したがって，従来的なデータベースを構築する際は，ストレージ領域に加えて，たくさんの CPU コアを搭載したマシンが用いられることが多い． また，データベースが巨大な場合は複数マシンにまたがった分散型のシステムが設計される． 分散型システムの場合は， Serverful クラウド (従来型) で議論したようにデータベースへのアクセス負荷に応じて適切なスケーリングがなされる必要がある．
DynamoDB は， AWS が提供しているサーバーレスな分散型データベースである． サーバーレスであるので，占有されたデータベース用仮想インスタンスは存在せず， API を通じてデータの書き込み・読み出し・検索などの操作を行う． S3 と同様に，データ保存領域の上限は定められておらず，データを入れれば入れた分だけ，保存領域は拡大していく． また，データベースへの負荷が増減したときのスケーリングは， DynamoDB が自動で行うので，ユーザーは心配する必要はない．
DynamoDB での利用料金の計算はやや複雑なのだが， "On-demand Capacity" というモードで使用した場合の料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 詳細は 公式ドキュメンテーション "Pricing for On-Demand Capacity" を参照)．
DynamoDB の利用料金
項目料金
Write request units
$1.25 per million write request units
Read request units
$0.25 per million read request units
Data storage
$0.25 per GB-month
DynamoDB ではデータの書き込み操作の単位を write request unit とよび，データの読み込み操作の単位を read request unit とよぶ． 基本的に， 1kB 以下のデータを一度書き込むと 1 write request unit を消費し，4kB 以下のデータを一度読み込むと 1 read request unit を消費する (詳しくは 公式ドキュメンテーション "Read/Write Capacity Mode" を参照のこと)． write request units は 100 万回ごとに $1.25, read request units は 100 万回ごとに $0.25 のコストが設定されている． また，保存されたデータ容量に対して $0.25 per GB のコストが月ごとに発生する． DynamoDB は高速な検索機能などを備えたデータベースであるので， GB あたりのストレージコストは S3 に比べ 10 倍程度高い． DynamoDB のデータの転送に関わるコストは，同じリージョン内ならば data-in，data-out ともに $0 である． リージョンをまたいだ通信には別途コストが発生する．
その他のサーバーレスクラウドの構成要素 ​
以上で紹介した Lambda, S3, DynamoDB がサーバーレスクラウドの中で最も使用する頻度が高いサービスになる． その他のサーバーレスクラウドの構成要素を以下に列挙する． いくつかについては，今後のハンズオンを行う中で改めて解説を行う．
  • API Gateway: API を構築する際のルーティングを担う． ??? で取り上げる．
  • Fargate: ??? で触れた Fargate も，サーバーレスクラウドの要素の一部である． Lambda との違いは，Lambda よりも大容量のメモリーや CPU を要するような計算などを行うことができる点が挙げられる．
  • Simple Notification Service (SNS): サーバーレスのサービス間でイベントをやり取りするためのサービス．
  • Step Functions: サーバーレスのサービス間のオーケストレーションを担う．
サーバーレスアーキテクチャは万能か？
この問いへの答えは，筆者は NO であると考える．
ここまで，サーバーレスの利点を強調して説明をしてきたが，まだまだ新しい技術なだけに，欠点，あるいはサーバーフルなシステムに劣る点は数多くある．
大きな欠点を一つあげるとすれば，サーバーレスのシステムは各クラウドプラットフォームに固有なものなので，特定のプラットフォームでしか運用できないシステムになってしまう点であろう． AWS で作成したサーバーレスのシステムを， Google のクラウドに移植するには，かなり大掛かりなプログラムの書き換えが必要になる． 一方， serverful なシステムであれば，プラットフォーム間のマイグレーションは比較的簡単に行うことができる． クラウドプロバイダーとしては，自社のシステムへの依存度を強めることで，顧客を離さないようにするという狙いがあるのだろう…
その他，サーバーレスコンピューティングの欠点や今後の課題などは，次の論文で詳しく議論されている． 興味のある読者はぜひ読んでいただきたい．
',56),c=[i];function h(m,g,u,b,f,_){return p(),n("div",null,c)}const y=s(d,[["render",h]]);export{q as __pageData,y as default};
diff --git a/assets/development_aws_serverless.md.7d293d5a.lean.js b/assets/development_aws_serverless.md.954dd7a1.lean.js
similarity index 88%
rename from assets/development_aws_serverless.md.7d293d5a.lean.js
rename to assets/development_aws_serverless.md.954dd7a1.lean.js
index d4dfedff..8216284c 100644
--- a/assets/development_aws_serverless.md.7d293d5a.lean.js
+++ b/assets/development_aws_serverless.md.954dd7a1.lean.js
@@ -1 +1 @@
-import{_ as e,a,b as t,c as r,d as o}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as s,c as n,o as p,R as l}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Serverless architecture","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/serverless.md","filePath":"development/aws/serverless.md","lastUpdated":1684858945000}'),d={name:"development/aws/serverless.md"},i=l("",56),c=[i];function h(m,g,u,b,f,_){return p(),n("div",null,c)}const y=s(d,[["render",h]]);export{q as __pageData,y as default};
+import{_ as e,a,b as t,c as r,d as o}from"./chunks/s3_vs_filesystem.fd65005d.js";import{_ as s,c as n,o as p,R as l}from"./chunks/framework.a1bac067.js";const q=JSON.parse('{"title":"Serverless architecture","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/serverless.md","filePath":"development/aws/serverless.md","lastUpdated":1684859551000}'),d={name:"development/aws/serverless.md"},i=l("",56),c=[i];function h(m,g,u,b,f,_){return p(),n("div",null,c)}const y=s(d,[["render",h]]);export{q as __pageData,y as default};
diff --git a/assets/development_aws_webserver.md.ab41d017.js b/assets/development_aws_webserver.md.a8b089c4.js
similarity index 99%
rename from assets/development_aws_webserver.md.ab41d017.js
rename to assets/development_aws_webserver.md.a8b089c4.js
index 85edbb6f..e4f8fd8b 100644
--- a/assets/development_aws_webserver.md.ab41d017.js
+++ b/assets/development_aws_webserver.md.a8b089c4.js
@@ -1 +1 @@
-import{_ as t,a as e}from"./chunks/rest_api.602a6e96.js";import{_ as o,c as d,o as r,R as a}from"./chunks/framework.a1bac067.js";const I=JSON.parse('{"title":"Web サービスの作り方","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/webserver.md","filePath":"development/aws/webserver.md","lastUpdated":1684858945000}'),s={name:"development/aws/webserver.md"},c=a('
Web サービスの作り方 ​
ここからが，本書第三部の内容になる． これまでのセクションでは，仮想サーバーをクラウド上に起動し，そこで計算を走らせる方法について解説をしてきた． EC2, ECS, Fargate, Batch などを利用して，動的にスケールするクラスターを構成し，並列にタスクを実行するクラウドシステムを実装してきた． 振り返ると，これまで紹介してきた内容は，自分自身が行いたい計算をクラウドを駆使することで実現する，という用途にフォーカスしていたことに気がつくだろう． 一方で，広く一般の人々に使ってもらえるような計算サービス・データベースを提供する，というのもクラウドの重要な役割として挙げられる．
本章から始まる第三部では，前回までとは少し方向性を変え，どのようにしてクラウド上にアプリケーションを展開し，広く一般の人に使ってもらうか，という点を講義したいと思う． これを通じて，どのようにして世の中のウェブサービスができ上がっているのかを知り，さらにどうやって自分でそのようなアプリケーションをゼロから構築するのか，という点を学んでもらう． その過程で，サーバーレスアーキテクチャという最新のクラウド設計手法を解説する．
その前準備として，本章ではどのようにしてウェブサービスが出来上がっているのか，その背後にある技術の概要を解説する． 用語の解説が中心となるが，後のハンズオンを実装するために必須の知識であるので，理解して前に進むよう心がけよう．
ウェブサービスの仕組み  — Twitter を例に ​
あなたがパソコンやスマートフォンから Twitter, Facebook, YouTube などのウェブサービスにアクセスしたとき，実際にどのようなことが行われ，コンテンツが提示されているのだろうか？
HTTP を通じたサーバーとクライアントのデータのやり取りは，すでに知っている読者も多いだろうし，逆にすべて解説しようとすると紙面が足りないので，ここではエッセンスの説明のみにとどめる． 以降では Twitter を具体例として，背後にあるサーバーとクライアントの間の通信を概説しよう． 概念図としては figure_title のような通信がクライアントとサーバーの間で行われていることになる．
クライアントと Web サーバーの通信の概念図
前提として，クライアントとサーバーの通信は HTTP (Hypertext Transfer Protocol) を使って行われる． また，最近では，暗号化された HTTP である HTTPS (HTTPS (Hypertext Transfer Protocol Secure)) を用いることがスタンダードになってきている． 第一のステップとして，クライアントは HTTP(S) 通信によってサーバーから静的なコンテンツを取得する． 静的なコンテンツとは， HTML (Hyptertext Markup Language) で記述されたウェブページの文書本体， CSS (Cascading Style Sheets) で記述されたページのデザインやレイアウトファイル，そして JavaScript (JS) で記述されたページの動的な挙動を定義したプログラム，が含まれる． Twitter を含む現代的なウェブアプリケーションの設計では，この静的なファイル群はページの”枠”を定義するだけで，中身となるコンテンツ (例: ツイートの一覧) は別途 API (Application Programming Interface) によって取得されなければならない． そこで，クライアントは先のステップで取得された JavaScript で定義されたプログラムに従って，サーバーに API を送信し，ツイートや画像データを取得する． この際，テキストデータのやり取りには JSON (JavaScript Object Notation) というフォーマットが用いられることが多い． 画像や動画などのコンテンツも同様に API により取得される． このようにして取得されたテキストや画像が，HTML の文書に埋め込まれることで，最終的にユーザーに提示されるページが完成するのである． また，新しいツイートを投稿するときにも，クライアントから API を通じてサーバーのデータベースにデータが書き込まれる．
REST API ​
API (Application Programming Interface) とはこれまで何度も出てきた言葉であるが，ここではよりフォーマルな定義付けを行う． API とはあるソフトウェア・アプリケーションが，外部のソフトウェアに対してコマンドやデータをやり取りするための媒介の一般的総称である． とくに，ウェブサービスの文脈では，サーバーが外界に対して提示しているコマンドの一覧のことを意味する． クライアントは，提示されている API から適切なコマンドを使うことによって，所望のデータを取得したり，あるいはサーバーにデータを送信したりする．
とくに，ウェブの文脈では REST (Representational State Transfer) とよばれる設計思想に基づいた API が現在では最も一般的に使われている． REST の設計指針に従った API のことを REST API あるいは RESTful API とよんだりする．
REST API は， figure_title に示したような Method と URI (Universal Resource Identifier) の組からなる．
REST API
Method (メソッド) とは，どのような操作を行いたいかを抽象的に表す，"動詞" として捉えることができる． メソッドには HTTP 規格で定義された 9 個の動詞 (verb) を使用することができる． この中でも， GET, POST, PUT, PATCH, DELETE の 5 個が最も頻繁に使用される (table_title)． この 5 つのメソッドによる操作を総称して CRUD (create, read, update, and delete) とよぶ．
REST API Methods
メソッド意図される動作
GET
要素を取得する
POST
新しい要素を作成する
PUT
既存の要素を新しい要素と置き換える
PATCH
既存の要素の一部を更新する
DELETE
要素を削除する
一方， URI は操作が行われる対象，すなわち "目的語" を表す． ウェブの文脈では操作が行われる対象のことをしばしば リソース とよぶ． URI は多くの場合 http または https から始まるウェブサーバーのアドレスから始まり， / (スラッシュ) 以降に所望のリソースのパスが指定される． figure_title の例で言えば， https://api.twitter.com というアドレスの /1.1/status/home_timeline というリソースを取得 (GET) せよ，という意味になる (なお，ここで 1.1 という数字は API のバージョンを示している)． この API リクエストによって，ユーザーのホームのタイムラインのツイートの一覧が取得される．
REST API のメソッドには， table_title で挙げたもの以外に， HTTP プロトコルで定義されているほかのメソッド (OPTIONS, TRACE など) を用いることもできるが，あまり一般的ではない．
また，これらのメソッドだけでは動詞として表現しきれないこともあるが， URI の名前でより意味を明確にすることもある． メソッドの使い方も，要素を削除する際は必ず DELETE を使わなければならない，という決まりもなく，たとえば， Twitter API でツイートを消す API は POST statuses/destroy/:id で定義されている． 最終的には，各ウェブサービスが公開している API ドキュメンテーションを読んで，それぞれの API がどんな操作をするのかを調べる必要がある．
REST の概念は 2000 年代初頭に確立され，今日の API 設計のスタンダードとなった． 一方で，ウェブのテクノロジーが進歩するにつれて，新たな API の設計アプローチの需要も高まっている． 近年とくに人気を集めているのが， GraphQL と呼ばれる API の設計方法である． GraphQL は Facebook 社によって最初に作られ，現在は GraghQL Foundation によって維持と更新がされている． GraphQL を使用すると，クライアントは REST と比較してより柔軟性の高いデータのクエリを行うことができるなど，いくつかの利点がある． キーワードだけでも知っておくと，今後役に立つだろう．
Twitter API ​
もう少し具体的にウェブサービスの API を体験する目的で，ここでは Twitter の API を見てみよう． Twitter が提供している API の一覧は Twitter の Developer Documentation で見ることができる． いくつかの代表的な API を table_title にまとめた．
Twitter API
エンドポイント動作
GET statuses/home_timeline
ホームのタイムラインのツイートの一覧を取得する．
GET statuses/show/:id
:id で指定されたツイートの詳細情報を取得する．
GET search
ツイートの検索を実行する．
POST statuses/update
新しいツイートを投稿する．
POST media/upload
画像をアップロードする
POST statuses/destroy/:id
:id で指定されたツイートを削除する．
POST statuses/retweet/:id
:id で指定されたツイートをリツイートする．
POST statuses/unretweet/:id
:id で指定されたツイートのリツイートを取り消す．
POST favorites/create
選択したツイートを"いいね"する．
POST favorites/destroy
選択したツイートを"いいね"を取り消す．
この API リストをもとに， Twitter のアプリまたはウェブサイトを開いたときに起こるクライアントとサーバーの通信をシミュレートしてみよう．
ユーザーが Twitter を開くと，まず最初に GET statuses/home_timeline の API リクエストによって，ユーザーのホームのタイムラインのツイートのリストが取得される． 個々のツイートは JSON 形式のデータになっており， id, text, user, coordinates, entities などの属性を含む． id はツイートに固有な ID を表し， text はツイートの本文を含んでいる． user はツイートを投稿したユーザーの名前やプロフィール画像の URL などを含んだ JSON データになっている． coordinates にはツイートが発信された地理的な座標が記録されている． また， entities にはツイートに関連するメディアファイル (画像など) のリンクなどの情報が埋め込まれている． GET statuses/home_timeline からは直近のツイートのリスト (リストが長すぎる場合は途中で切られたもの) が取得される． もしツイートの ID を知っている場合は GET statuses/show/:id を呼ぶことによって， :id パラメータで指定された特定のツイートを取得することができる．
ツイートの検索を行うためには GET search API を使用する． この API には，ツイートに含まれる単語や，ハッシュタグ，ツイートの発信された日時や場所など，様々なクエリの条件を渡すことができる． API からは， GET statuses/home_timeline などと同様， JSON 形式のツイートのデータが返される．
ユーザーが新しいツイートを投稿するには POST statuses/update のエンドポイントを利用する． POST statuses/update には，ツイートの文章や，リプライの場合はリプライ先のツイートの ID などのデータを送信する． また，ツイートに画像データを添付したい場合は， POST media/upload を併せて使用する． ツイートの削除を行うには， POST statuses/destroy/:id を用いる．
そのほか，頻繁に行われる操作としては， POST statuses/retweet/:id と POST statuses/unretweet/:id がある． これらは， :id で指定されるツイートに対して，それぞれリツイートを実行あるいは取り消すための API である． また， POST favorites/create，POST favorites/destroy を使用することによって，選択されたツイートに"いいね"を追加したり，取り消したりする操作を行う．
このような一連の操作が， Twitter のアプリの背後では行われている． また，自分自身でボットを作成したい場合は，これらの API を適切に組み合わせ，カスタムのプログラムを書くことで実現される．
このように， API はあらゆるウェブサービスを作るうえで一番基礎となる要素である． 次からの章では本章で紹介した用語が何度も出てくるので，頭の片隅に置いたうえで読み進めていただきたい．
',30),p=[c];function i(n,l,T,P,h,u){return r(),d("div",null,p)}const S=o(s,[["render",i]]);export{I as __pageData,S as default};
+import{_ as t,a as e}from"./chunks/rest_api.602a6e96.js";import{_ as o,c as d,o as r,R as a}from"./chunks/framework.a1bac067.js";const I=JSON.parse('{"title":"Web サービスの作り方","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/webserver.md","filePath":"development/aws/webserver.md","lastUpdated":1684859551000}'),s={name:"development/aws/webserver.md"},c=a('
Web サービスの作り方 ​
ここからが，本書第三部の内容になる． これまでのセクションでは，仮想サーバーをクラウド上に起動し，そこで計算を走らせる方法について解説をしてきた． EC2, ECS, Fargate, Batch などを利用して，動的にスケールするクラスターを構成し，並列にタスクを実行するクラウドシステムを実装してきた． 振り返ると，これまで紹介してきた内容は，自分自身が行いたい計算をクラウドを駆使することで実現する，という用途にフォーカスしていたことに気がつくだろう． 一方で，広く一般の人々に使ってもらえるような計算サービス・データベースを提供する，というのもクラウドの重要な役割として挙げられる．
本章から始まる第三部では，前回までとは少し方向性を変え，どのようにしてクラウド上にアプリケーションを展開し，広く一般の人に使ってもらうか，という点を講義したいと思う． これを通じて，どのようにして世の中のウェブサービスができ上がっているのかを知り，さらにどうやって自分でそのようなアプリケーションをゼロから構築するのか，という点を学んでもらう． その過程で，サーバーレスアーキテクチャという最新のクラウド設計手法を解説する．
その前準備として，本章ではどのようにしてウェブサービスが出来上がっているのか，その背後にある技術の概要を解説する． 用語の解説が中心となるが，後のハンズオンを実装するために必須の知識であるので，理解して前に進むよう心がけよう．
ウェブサービスの仕組み  — Twitter を例に ​
あなたがパソコンやスマートフォンから Twitter, Facebook, YouTube などのウェブサービスにアクセスしたとき，実際にどのようなことが行われ，コンテンツが提示されているのだろうか？
HTTP を通じたサーバーとクライアントのデータのやり取りは，すでに知っている読者も多いだろうし，逆にすべて解説しようとすると紙面が足りないので，ここではエッセンスの説明のみにとどめる． 以降では Twitter を具体例として，背後にあるサーバーとクライアントの間の通信を概説しよう． 概念図としては figure_title のような通信がクライアントとサーバーの間で行われていることになる．
クライアントと Web サーバーの通信の概念図
前提として，クライアントとサーバーの通信は HTTP (Hypertext Transfer Protocol) を使って行われる． また，最近では，暗号化された HTTP である HTTPS (HTTPS (Hypertext Transfer Protocol Secure)) を用いることがスタンダードになってきている． 第一のステップとして，クライアントは HTTP(S) 通信によってサーバーから静的なコンテンツを取得する． 静的なコンテンツとは， HTML (Hyptertext Markup Language) で記述されたウェブページの文書本体， CSS (Cascading Style Sheets) で記述されたページのデザインやレイアウトファイル，そして JavaScript (JS) で記述されたページの動的な挙動を定義したプログラム，が含まれる． Twitter を含む現代的なウェブアプリケーションの設計では，この静的なファイル群はページの”枠”を定義するだけで，中身となるコンテンツ (例: ツイートの一覧) は別途 API (Application Programming Interface) によって取得されなければならない． そこで，クライアントは先のステップで取得された JavaScript で定義されたプログラムに従って，サーバーに API を送信し，ツイートや画像データを取得する． この際，テキストデータのやり取りには JSON (JavaScript Object Notation) というフォーマットが用いられることが多い． 画像や動画などのコンテンツも同様に API により取得される． このようにして取得されたテキストや画像が，HTML の文書に埋め込まれることで，最終的にユーザーに提示されるページが完成するのである． また，新しいツイートを投稿するときにも，クライアントから API を通じてサーバーのデータベースにデータが書き込まれる．
REST API ​
API (Application Programming Interface) とはこれまで何度も出てきた言葉であるが，ここではよりフォーマルな定義付けを行う． API とはあるソフトウェア・アプリケーションが，外部のソフトウェアに対してコマンドやデータをやり取りするための媒介の一般的総称である． とくに，ウェブサービスの文脈では，サーバーが外界に対して提示しているコマンドの一覧のことを意味する． クライアントは，提示されている API から適切なコマンドを使うことによって，所望のデータを取得したり，あるいはサーバーにデータを送信したりする．
とくに，ウェブの文脈では REST (Representational State Transfer) とよばれる設計思想に基づいた API が現在では最も一般的に使われている． REST の設計指針に従った API のことを REST API あるいは RESTful API とよんだりする．
REST API は， figure_title に示したような Method と URI (Universal Resource Identifier) の組からなる．
REST API
Method (メソッド) とは，どのような操作を行いたいかを抽象的に表す，"動詞" として捉えることができる． メソッドには HTTP 規格で定義された 9 個の動詞 (verb) を使用することができる． この中でも， GET, POST, PUT, PATCH, DELETE の 5 個が最も頻繁に使用される (table_title)． この 5 つのメソッドによる操作を総称して CRUD (create, read, update, and delete) とよぶ．
REST API Methods
メソッド意図される動作
GET
要素を取得する
POST
新しい要素を作成する
PUT
既存の要素を新しい要素と置き換える
PATCH
既存の要素の一部を更新する
DELETE
要素を削除する
一方， URI は操作が行われる対象，すなわち "目的語" を表す． ウェブの文脈では操作が行われる対象のことをしばしば リソース とよぶ． URI は多くの場合 http または https から始まるウェブサーバーのアドレスから始まり， / (スラッシュ) 以降に所望のリソースのパスが指定される． figure_title の例で言えば， https://api.twitter.com というアドレスの /1.1/status/home_timeline というリソースを取得 (GET) せよ，という意味になる (なお，ここで 1.1 という数字は API のバージョンを示している)． この API リクエストによって，ユーザーのホームのタイムラインのツイートの一覧が取得される．
REST API のメソッドには， table_title で挙げたもの以外に， HTTP プロトコルで定義されているほかのメソッド (OPTIONS, TRACE など) を用いることもできるが，あまり一般的ではない．
また，これらのメソッドだけでは動詞として表現しきれないこともあるが， URI の名前でより意味を明確にすることもある． メソッドの使い方も，要素を削除する際は必ず DELETE を使わなければならない，という決まりもなく，たとえば， Twitter API でツイートを消す API は POST statuses/destroy/:id で定義されている． 最終的には，各ウェブサービスが公開している API ドキュメンテーションを読んで，それぞれの API がどんな操作をするのかを調べる必要がある．
REST の概念は 2000 年代初頭に確立され，今日の API 設計のスタンダードとなった． 一方で，ウェブのテクノロジーが進歩するにつれて，新たな API の設計アプローチの需要も高まっている． 近年とくに人気を集めているのが， GraphQL と呼ばれる API の設計方法である． GraphQL は Facebook 社によって最初に作られ，現在は GraghQL Foundation によって維持と更新がされている． GraphQL を使用すると，クライアントは REST と比較してより柔軟性の高いデータのクエリを行うことができるなど，いくつかの利点がある． キーワードだけでも知っておくと，今後役に立つだろう．
Twitter API ​
もう少し具体的にウェブサービスの API を体験する目的で，ここでは Twitter の API を見てみよう． Twitter が提供している API の一覧は Twitter の Developer Documentation で見ることができる． いくつかの代表的な API を table_title にまとめた．
Twitter API
エンドポイント動作
GET statuses/home_timeline
ホームのタイムラインのツイートの一覧を取得する．
GET statuses/show/:id
:id で指定されたツイートの詳細情報を取得する．
GET search
ツイートの検索を実行する．
POST statuses/update
新しいツイートを投稿する．
POST media/upload
画像をアップロードする
POST statuses/destroy/:id
:id で指定されたツイートを削除する．
POST statuses/retweet/:id
:id で指定されたツイートをリツイートする．
POST statuses/unretweet/:id
:id で指定されたツイートのリツイートを取り消す．
POST favorites/create
選択したツイートを"いいね"する．
POST favorites/destroy
選択したツイートを"いいね"を取り消す．
この API リストをもとに， Twitter のアプリまたはウェブサイトを開いたときに起こるクライアントとサーバーの通信をシミュレートしてみよう．
ユーザーが Twitter を開くと，まず最初に GET statuses/home_timeline の API リクエストによって，ユーザーのホームのタイムラインのツイートのリストが取得される． 個々のツイートは JSON 形式のデータになっており， id, text, user, coordinates, entities などの属性を含む． id はツイートに固有な ID を表し， text はツイートの本文を含んでいる． user はツイートを投稿したユーザーの名前やプロフィール画像の URL などを含んだ JSON データになっている． coordinates にはツイートが発信された地理的な座標が記録されている． また， entities にはツイートに関連するメディアファイル (画像など) のリンクなどの情報が埋め込まれている． GET statuses/home_timeline からは直近のツイートのリスト (リストが長すぎる場合は途中で切られたもの) が取得される． もしツイートの ID を知っている場合は GET statuses/show/:id を呼ぶことによって， :id パラメータで指定された特定のツイートを取得することができる．
ツイートの検索を行うためには GET search API を使用する． この API には，ツイートに含まれる単語や，ハッシュタグ，ツイートの発信された日時や場所など，様々なクエリの条件を渡すことができる． API からは， GET statuses/home_timeline などと同様， JSON 形式のツイートのデータが返される．
ユーザーが新しいツイートを投稿するには POST statuses/update のエンドポイントを利用する． POST statuses/update には，ツイートの文章や，リプライの場合はリプライ先のツイートの ID などのデータを送信する． また，ツイートに画像データを添付したい場合は， POST media/upload を併せて使用する． ツイートの削除を行うには， POST statuses/destroy/:id を用いる．
そのほか，頻繁に行われる操作としては， POST statuses/retweet/:id と POST statuses/unretweet/:id がある． これらは， :id で指定されるツイートに対して，それぞれリツイートを実行あるいは取り消すための API である． また， POST favorites/create，POST favorites/destroy を使用することによって，選択されたツイートに"いいね"を追加したり，取り消したりする操作を行う．
このような一連の操作が， Twitter のアプリの背後では行われている． また，自分自身でボットを作成したい場合は，これらの API を適切に組み合わせ，カスタムのプログラムを書くことで実現される．
このように， API はあらゆるウェブサービスを作るうえで一番基礎となる要素である． 次からの章では本章で紹介した用語が何度も出てくるので，頭の片隅に置いたうえで読み進めていただきたい．
',30),p=[c];function i(n,l,T,P,h,u){return r(),d("div",null,p)}const S=o(s,[["render",i]]);export{I as __pageData,S as default};
diff --git a/assets/development_aws_webserver.md.ab41d017.lean.js b/assets/development_aws_webserver.md.a8b089c4.lean.js
similarity index 87%
rename from assets/development_aws_webserver.md.ab41d017.lean.js
rename to assets/development_aws_webserver.md.a8b089c4.lean.js
index 73f357af..f01ad989 100644
--- a/assets/development_aws_webserver.md.ab41d017.lean.js
+++ b/assets/development_aws_webserver.md.a8b089c4.lean.js
@@ -1 +1 @@
-import{_ as t,a as e}from"./chunks/rest_api.602a6e96.js";import{_ as o,c as d,o as r,R as a}from"./chunks/framework.a1bac067.js";const I=JSON.parse('{"title":"Web サービスの作り方","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/webserver.md","filePath":"development/aws/webserver.md","lastUpdated":1684858945000}'),s={name:"development/aws/webserver.md"},c=a("",30),p=[c];function i(n,l,T,P,h,u){return r(),d("div",null,p)}const S=o(s,[["render",i]]);export{I as __pageData,S as default};
+import{_ as t,a as e}from"./chunks/rest_api.602a6e96.js";import{_ as o,c as d,o as r,R as a}from"./chunks/framework.a1bac067.js";const I=JSON.parse('{"title":"Web サービスの作り方","description":"","frontmatter":{},"headers":[],"relativePath":"development/aws/webserver.md","filePath":"development/aws/webserver.md","lastUpdated":1684859551000}'),s={name:"development/aws/webserver.md"},c=a("",30),p=[c];function i(n,l,T,P,h,u){return r(),d("div",null,p)}const S=o(s,[["render",i]]);export{I as __pageData,S as default};
diff --git a/assets/development_file-naming-convention.md.404e82de.js b/assets/development_file-naming-convention.md.404e82de.js
new file mode 100644
index 00000000..38d50ae3
--- /dev/null
+++ b/assets/development_file-naming-convention.md.404e82de.js
@@ -0,0 +1,67 @@
+import{_ as e,c as s,o as a,R as n}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"File Name Conventions","description":"","frontmatter":{},"headers":[],"relativePath":"development/file-naming-convention.md","filePath":"development/file-naming-convention.md","lastUpdated":1684859551000}'),i={name:"development/file-naming-convention.md"},l=n(`
File Name Conventions ​
A collection of guidelines for writing file names used in web projects.
Possible characters ​
Use dashes as delimiters ​
  • You should use dashes (-) as delimiters.
  • Periods are allowed in some cases, such as for languages and conditions.
  • Never use spaces or underscores. Spaces are converted to %20 in URLs or can break an URL when shared. Underscores are difficult to see when the file name is displayed as an underlined link. Although the use of underscores does not impact your ranking that much, Google advices not to use underscores.
  • Exceptions apply for functional requirements, such as for Sass partials. A leading underscore informs the Sass compiler a file is only a partial file and should never be generated into a stand alone CSS file.
Right:
file-name-with-dashes.en.min.html
file-name-with-dashes.en.min.html
Do not use special characters ​
Avoid using non-alphanumeric characters in file names, such as: '*' ':' '\\' '/' '<' '>' '|' '"' '!' '?' '[' ']' ';' '=' '+' '&' '£' '$' '€' '%' or ','. These characters can have special meaning in programming languages or can cause problems with different operating systems.
Use lowercase, never uppercase ​
We should always consider URLs as case-sensitive according to W3.org. Therefore, use lowercase to reduces errors when typing URLs.
Write sections of a file name in a consistent order ​
Sections should be written in this order:
  1. Description
  2. Number
  3. Date
  4. Target device, image size, pixel density
  5. Version number
  6. Status
  7. Language code
  8. File conditions
Possible combinations
description-01-20150102-palm-1.0b-draft.en.min.js /* extreme combination */
+description.min.js
+description.en.html
+description-01.jpg
+description-02.jpg
+description-1024x768_2x.jpg
+description-desk_2x.jpg
description-01-20150102-palm-1.0b-draft.en.min.js /* extreme combination */
+description.min.js
+description.en.html
+description-01.jpg
+description-02.jpg
+description-1024x768_2x.jpg
+description-desk_2x.jpg
Write description for developers and users ​
Don't be afraid to write long informative file names. Few people type a file name manually and most operating systems support 255 characters. But only add information that makes it easy for users and developers to recognize files from one another at a glance. For the description use information such as:
  • type of data
  • project name or acronym
  • subjects
  • people's names
  • characteristics
  • location
Give your images detailed, informative filenames The filename can give Google clues about the subject matter of the image. Try to make your filename a good description of the subject matter of the image. For example, my-new-black-kitten.jpg is a lot more informative than IMG00023.JPG. Descriptive filenames can also be useful to users: If we're unable to find suitable text in the page on which we found the image, we'll use the filename as the image's snippet in our search results. - Google, 2015, Image publishing guidelines
Keep people's names compact ​
Sometimes you want to include the name of a person in the file name, e.g. the author or the person in the photo.
  • Write names without word delimiters.
  • Only write initials for the first name and write the last name in full. There are two exceptions: (1) when the last name or final file name is very long you may use initials for the last name; (2) when there is already a person with the same combination, you may write the first name in full.
  • If people have exactly the same name written in full, you can add a number: firstnamelastname2.
Right:
bvandebiezen.jpg
+shoogenhout.jpg
+a-very-long-description-with-name-bvdb.jpg
+asmith.jpg
+adamsmith.jpg
+adamsmith2.jpg
bvandebiezen.jpg
+shoogenhout.jpg
+a-very-long-description-with-name-bvdb.jpg
+asmith.jpg
+adamsmith.jpg
+adamsmith2.jpg
Use two or more digits to distinguish sequential files with the same description ​
  • Start number with a dash as a delimiter.
  • Add a zero to single digit numbers, e.g. '01' instead of '1'.
  • Numbers may also be placed before the description if needed. A dash will still be used as delimiter with the description.
Right:
description-01.jpg
+description-02.jpg
+description-03.jpg
+description-04.jpg
+
+01-description.jpg
+02-description.jpg
+03-description.jpg
+04-description.jpg
description-01.jpg
+description-02.jpg
+description-03.jpg
+description-04.jpg
+
+01-description.jpg
+02-description.jpg
+03-description.jpg
+04-description.jpg
Keep dates or date ranges compact and start with year ​
  • Write dates without delimiters.
  • Always use four digits for years, two digits for months and two digits for days.
  • Start with year, then month, and end with day if available: yyyymmdd, yyyymm, or yyyy. This makes sure similar file names are sorted by date when sorted alphabetically.
  • Use a double dash to separate two dates describing an interval: yyyy--yyyy. Start with the earliest date.
Right:
description-20150401.php
+description-201504.php
+description-2015.php
+description-2000--2010.php
description-20150401.php
+description-201504.php
+description-2015.php
+description-2000--2010.php
See 'ISO 8601' for further reading.
Use special modifiers for target devices, image sizes or media queries, and pixel densities. ​
Modifiers are inspired by Apple iOS naming conventions. There are some differences. Apple uses '@' as a delimiter for the section indicating higher resolution images, for example '@2x' for retina images. Because '@' is a reserved character and can create problems, we use Bourbon's convention: an underscore. Also, Apple uses a tilde (~) as a delimiter for a section indicating specific devices. Because also a tilde can create problems, we suggest to simply use a dash.
  • Order should be: (1) target device or media query, (2) size, (3) pixel density.
  • Start target device or media queries with a dash (-) as delimiter.
  • Start image sizes with a dash (-) as delimiter.
  • Start pixel density with an underscore (_) as delimiter, for example '_2x' or '_3x'.
  • When only a width or height is available or applicable, add a 'w' for width or 'h' for height directly after the the amount of pixels.
  • When both measurements are available, do not add a 'w' or 'h' and separate the width and height with an 'x'.
  • When both the width and height should not exceed a dimension but the images should keep the original aspect ratio, add a 'max' (maximum) after the amount of pixels.
Right:
description_2x.jpg
+description-lap.jpg
+description-desk.jpg
+description-lap_2x.jpg
+description-palm-1024w_2x.jpg
+description-iphone5-568h_2x.jpg
+description-palm-1024x768_2x.jpg
+description-40max.jpg
description_2x.jpg
+description-lap.jpg
+description-desk.jpg
+description-lap_2x.jpg
+description-palm-1024w_2x.jpg
+description-iphone5-568h_2x.jpg
+description-palm-1024x768_2x.jpg
+description-40max.jpg
Use version numbers if available ​
  • Start version with a dash (-) as delimiter.
  • Use periods (.) to separate point releases.
  • Always add trailing zeros to major releases, e.g. '2.0' instead of '2'.
  • Types, such as 'a' (alpha), 'b' (beta), 'rc1' (release candidate 1) can be added without delimiters.
Right:
description-0.5.js
+description-1.0b.js
+description-1.0rc1.js
description-0.5.js
+description-1.0b.js
+description-1.0rc1.js
Add status when needed ​
  • You can optionally add a file status such as 'draft' and 'published'.
  • Start status with a dash.
Right:
description-draft.md
description-draft.md
Add language code only when different languages are available ​
  • Use a period to separate the language code from the rest of the file name.
  • Use ISO 639-1, two-letter codes, for languages.
  • Only add languages when different languages are available.
Right:
description.nl.txt
+description.en.txt
description.nl.txt
+description.en.txt
Add file conditions just before the file extension ​
  • The file condition should be the last part, just before the file extension.
  • Use a period (.) to separate the condition from the rest of the file name.
  • Use periods (.) as a delimiter for different conditions.
Right:
description.min.js
+description.custom1234.min.js
description.min.js
+description.custom1234.min.js
Rewrite original file names not following conventions ​
It is not preferred to keep file names in it's original format if it doesn't match your file name conventions. But in some cases it is easier to keep the file name untouched. Sometimes you want to easily replace a file with a newer one from the original source in the future.
`,57),o=[l];function t(r,p,c,d,h,b){return a(),s("div",null,o)}const g=e(i,[["render",t]]);export{u as __pageData,g as default};
diff --git a/assets/development_file-naming-convention.md.404e82de.lean.js b/assets/development_file-naming-convention.md.404e82de.lean.js
new file mode 100644
index 00000000..e23f2d9f
--- /dev/null
+++ b/assets/development_file-naming-convention.md.404e82de.lean.js
@@ -0,0 +1 @@
+import{_ as e,c as s,o as a,R as n}from"./chunks/framework.a1bac067.js";const u=JSON.parse('{"title":"File Name Conventions","description":"","frontmatter":{},"headers":[],"relativePath":"development/file-naming-convention.md","filePath":"development/file-naming-convention.md","lastUpdated":1684859551000}'),i={name:"development/file-naming-convention.md"},l=n("",57),o=[l];function t(r,p,c,d,h,b){return a(),s("div",null,o)}const g=e(i,[["render",t]]);export{u as __pageData,g as default};
diff --git a/assets/development_rclone-for-r2.md.e7f3a9b7.js b/assets/development_rclone-for-r2.md.e7f3a9b7.js
new file mode 100644
index 00000000..8c5b617e
--- /dev/null
+++ b/assets/development_rclone-for-r2.md.e7f3a9b7.js
@@ -0,0 +1,55 @@
+import{_ as s,c as n,o as a,R as e}from"./chunks/framework.a1bac067.js";const B=JSON.parse('{"title":"Configure rclone for R2","description":"","frontmatter":{},"headers":[],"relativePath":"development/rclone-for-r2.md","filePath":"development/rclone-for-r2.md","lastUpdated":1684859551000}'),l={name:"development/rclone-for-r2.md"},o=e(`
Configure rclone for R2 ​
Example of how to configure rclone to use R2.
You must generate an Access Key before getting started. All examples will utilize access_key_id and access_key_secret variables which represent the Access Key ID and Secret Access Key values you generated.
With [rclone](https://rclone.org/install/) installed, you may run [rclone config](https://rclone.org/s3/) to configure a new S3 storage provider. You will be prompted with a series of questions for the new prvider details.
If you have already configured rclone in the past, you may run rclone config file to print the location of your rclone configuration file:
bash
$ rclone config file
+# Configuration file is stored at:
+# ~/.config/rclone/rclone.conf
$ rclone config file
+# Configuration file is stored at:
+# ~/.config/rclone/rclone.conf
Then use an editor (nano or vim, for example) to add or edit the new provider. This example assumes you are adding a new r2demo provider:
bash
[r2demo]
+type = s3
+provider = Cloudflare
+access_key_id = abc123 # Your access_key_id
+secret_access_key = xyz456 # Your access_key_secret
+endpoint = https://<accountid>.r2.cloudflarestorage.com
+acl = private
[r2demo]
+type = s3
+provider = Cloudflare
+access_key_id = abc123 # Your access_key_id
+secret_access_key = xyz456 # Your access_key_secret
+endpoint = https://<accountid>.r2.cloudflarestorage.com
+acl = private
You may then use the new rclone provider for any of your normal workflows.
List buckets & objects ​
The rclone tree command can be used to list the contents of the remote, in this case Cloudflare R2.
bash
$ rclone tree r2demo:
+# /
+# ├── user-uploads
+# │   └── foobar.png
+# └── my-bucket-name
+#     ├── cat.png
+#     └── todos.txt
+
+$ rclone tree r2demo:my-bucket-name
+# /
+# ├── cat.png
+# └── todos.txt
$ rclone tree r2demo:
+# /
+# ├── user-uploads
+# │   └── foobar.png
+# └── my-bucket-name
+#     ├── cat.png
+#     └── todos.txt
+
+$ rclone tree r2demo:my-bucket-name
+# /
+# ├── cat.png
+# └── todos.txt
Upload and retrieve objects ​
The rclone copy command can be used to upload objects to an R2 bucket and vice versa - this allows you to upload files up to the 5 TB maximum object size that R2 supports.
bash
# Upload dog.txt to the user-uploads bucket
+$ rclone copy dog.txt r2demo:user-uploads/dog.txt
+$ rclone tree r2demo:user-uploads
+# /
+# ├── foobar.png
+# └── dog.txt
+
+# Download dog.txt from the user-uploads bucket
+$ rclone copy r2demo:user-uploads/dog.txt dog.txt
# Upload dog.txt to the user-uploads bucket
+$ rclone copy dog.txt r2demo:user-uploads/dog.txt
+$ rclone tree r2demo:user-uploads
+# /
+# ├── foobar.png
+# └── dog.txt
+
+# Download dog.txt from the user-uploads bucket
+$ rclone copy r2demo:user-uploads/dog.txt dog.txt
Generate presigned URLs ​
You can also generate presigned links which allow you to share public access to a file temporarily using the rclone link command.
`,17),p=[o];function t(r,c,i,y,d,C){return a(),n("div",null,p)}const b=s(l,[["render",t]]);export{B as __pageData,b as default};
diff --git a/assets/development_rclone-for-r2.md.e7f3a9b7.lean.js b/assets/development_rclone-for-r2.md.e7f3a9b7.lean.js
new file mode 100644
index 00000000..a3789294
--- /dev/null
+++ b/assets/development_rclone-for-r2.md.e7f3a9b7.lean.js
@@ -0,0 +1 @@
+import{_ as s,c as n,o as a,R as e}from"./chunks/framework.a1bac067.js";const B=JSON.parse('{"title":"Configure rclone for R2","description":"","frontmatter":{},"headers":[],"relativePath":"development/rclone-for-r2.md","filePath":"development/rclone-for-r2.md","lastUpdated":1684859551000}'),l={name:"development/rclone-for-r2.md"},o=e("",17),p=[o];function t(r,c,i,y,d,C){return a(),n("div",null,p)}const b=s(l,[["render",t]]);export{B as __pageData,b as default};
diff --git a/assets/getting-started.md.bd1ab99e.js b/assets/getting-started.md.ad0edf0e.js
similarity index 99%
rename from assets/getting-started.md.bd1ab99e.js
rename to assets/getting-started.md.ad0edf0e.js
index 3c34e709..09f6f209 100644
--- a/assets/getting-started.md.bd1ab99e.js
+++ b/assets/getting-started.md.ad0edf0e.js
@@ -1,4 +1,4 @@
-import{_ as s,c as a,o,R as n}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Git push results in \\"Authentication Failed”","description":"","frontmatter":{"title":"Git push results in \\"Authentication Failed”","author":"Anda Toshiki","date":"2023/2/21/ 10:54","editLink":true,"lastUpdated":true},"headers":[],"relativePath":"getting-started.md","filePath":"getting-started.md","lastUpdated":1684858945000}'),l={name:"getting-started.md"},e=n(`
Git push results in "Authentication Failed” ​
Issue ​
bash
$ git push
+import{_ as s,c as a,o,R as n}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Git push results in \\"Authentication Failed”","description":"","frontmatter":{"title":"Git push results in \\"Authentication Failed”","author":"Anda Toshiki","date":"2023/2/21/ 10:54","editLink":true,"lastUpdated":true},"headers":[],"relativePath":"getting-started.md","filePath":"getting-started.md","lastUpdated":1684859551000}'),l={name:"getting-started.md"},e=n(`
Git push results in "Authentication Failed” ​
Issue ​
bash
$ git push
 Username for 'https://github.com': \${{ GITHUB_USERNAME }}
 Password for 'https://andatoshiki@github.com': \${{ GITHUB_PASSWORD }} # or GitHub personal access token
 remote: Invalid username or password.
diff --git a/assets/getting-started.md.bd1ab99e.lean.js b/assets/getting-started.md.ad0edf0e.lean.js
similarity index 89%
rename from assets/getting-started.md.bd1ab99e.lean.js
rename to assets/getting-started.md.ad0edf0e.lean.js
index 98a413f4..ba37aed1 100644
--- a/assets/getting-started.md.bd1ab99e.lean.js
+++ b/assets/getting-started.md.ad0edf0e.lean.js
@@ -1 +1 @@
-import{_ as s,c as a,o,R as n}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Git push results in \\"Authentication Failed”","description":"","frontmatter":{"title":"Git push results in \\"Authentication Failed”","author":"Anda Toshiki","date":"2023/2/21/ 10:54","editLink":true,"lastUpdated":true},"headers":[],"relativePath":"getting-started.md","filePath":"getting-started.md","lastUpdated":1684858945000}'),l={name:"getting-started.md"},e=n("",15),t=[e];function p(r,c,i,y,u,C){return o(),a("div",null,t)}const h=s(l,[["render",p]]);export{A as __pageData,h as default};
+import{_ as s,c as a,o,R as n}from"./chunks/framework.a1bac067.js";const A=JSON.parse('{"title":"Git push results in \\"Authentication Failed”","description":"","frontmatter":{"title":"Git push results in \\"Authentication Failed”","author":"Anda Toshiki","date":"2023/2/21/ 10:54","editLink":true,"lastUpdated":true},"headers":[],"relativePath":"getting-started.md","filePath":"getting-started.md","lastUpdated":1684859551000}'),l={name:"getting-started.md"},e=n("",15),t=[e];function p(r,c,i,y,u,C){return o(),a("div",null,t)}const h=s(l,[["render",p]]);export{A as __pageData,h as default};
diff --git a/assets/index.md.6f6f33ad.js b/assets/index.md.4559d5f1.js
similarity index 83%
rename from assets/index.md.6f6f33ad.js
rename to assets/index.md.4559d5f1.js
index f0627542..cc04e13f 100644
--- a/assets/index.md.6f6f33ad.js
+++ b/assets/index.md.4559d5f1.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a}from"./chunks/framework.a1bac067.js";const c=JSON.parse(`{"title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content craetion and notes management.","description":"","frontmatter":{"layout":"home","title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content craetion and notes management.","hero":{"name":"Toshiki's Notebook","text":"Research & Produce","tagline":"👨‍💻 Eternal & digital knowledge base for content craetion and notes management.","image":{"src":"/logos/logo.svg","alt":"Home logo"},"actions":[{"theme":"brand","text":"👉 Start Reading","link":"/getting-started"},{"theme":"alt","text":"🗒️ View on GitHub","link":"https://github.com/andatoshiki/toshiki-notebook"}]},"features":[{"icon":"🕒","title":"Tempus Fugit","details":"Tempus Fugit, which means \\"time flies\\" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential."},{"icon":"☕","title":"Carpe Diem","details":"Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning."},{"icon":"💡","title":"Epiphania","details":"An epiphany, derived from the Latin word \\"epiphania,\\" is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas."}]},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":1684858945000}`),i={name:"index.md"};function n(o,s,r,l,h,d){return a(),t("div")}const p=e(i,[["render",n]]);export{c as __pageData,p as default};
+import{_ as e,c as t,o as a}from"./chunks/framework.a1bac067.js";const c=JSON.parse(`{"title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content creation and notes management.","description":"","frontmatter":{"layout":"home","title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content creation and notes management.","hero":{"name":"Toshiki's Notebook","text":"Research & Produce","tagline":"👨‍💻 Eternal & digital knowledge base for content creation and notes management.","image":{"src":"/logos/logo.svg","alt":"Home logo"},"actions":[{"theme":"brand","text":"👉 Start Reading","link":"/getting-started"},{"theme":"alt","text":"🗒️ View on GitHub","link":"https://github.com/andatoshiki/toshiki-notebook"}]},"features":[{"icon":"🕒","title":"Tempus Fugit","details":"Tempus Fugit, which means \\"time flies\\" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential."},{"icon":"☕","title":"Carpe Diem","details":"Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning."},{"icon":"💡","title":"Epiphania","details":"An epiphany, derived from the Latin word \\"epiphania,\\" is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas."}]},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":1684859551000}`),i={name:"index.md"};function n(o,s,r,l,d,h){return a(),t("div")}const p=e(i,[["render",n]]);export{c as __pageData,p as default};
diff --git a/assets/index.md.6f6f33ad.lean.js b/assets/index.md.4559d5f1.lean.js
similarity index 83%
rename from assets/index.md.6f6f33ad.lean.js
rename to assets/index.md.4559d5f1.lean.js
index f0627542..cc04e13f 100644
--- a/assets/index.md.6f6f33ad.lean.js
+++ b/assets/index.md.4559d5f1.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a}from"./chunks/framework.a1bac067.js";const c=JSON.parse(`{"title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content craetion and notes management.","description":"","frontmatter":{"layout":"home","title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content craetion and notes management.","hero":{"name":"Toshiki's Notebook","text":"Research & Produce","tagline":"👨‍💻 Eternal & digital knowledge base for content craetion and notes management.","image":{"src":"/logos/logo.svg","alt":"Home logo"},"actions":[{"theme":"brand","text":"👉 Start Reading","link":"/getting-started"},{"theme":"alt","text":"🗒️ View on GitHub","link":"https://github.com/andatoshiki/toshiki-notebook"}]},"features":[{"icon":"🕒","title":"Tempus Fugit","details":"Tempus Fugit, which means \\"time flies\\" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential."},{"icon":"☕","title":"Carpe Diem","details":"Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning."},{"icon":"💡","title":"Epiphania","details":"An epiphany, derived from the Latin word \\"epiphania,\\" is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas."}]},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":1684858945000}`),i={name:"index.md"};function n(o,s,r,l,h,d){return a(),t("div")}const p=e(i,[["render",n]]);export{c as __pageData,p as default};
+import{_ as e,c as t,o as a}from"./chunks/framework.a1bac067.js";const c=JSON.parse(`{"title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content creation and notes management.","description":"","frontmatter":{"layout":"home","title":"Toshiki's Notebook","titleTemplate":"Eternal digital knowledge base for content creation and notes management.","hero":{"name":"Toshiki's Notebook","text":"Research & Produce","tagline":"👨‍💻 Eternal & digital knowledge base for content creation and notes management.","image":{"src":"/logos/logo.svg","alt":"Home logo"},"actions":[{"theme":"brand","text":"👉 Start Reading","link":"/getting-started"},{"theme":"alt","text":"🗒️ View on GitHub","link":"https://github.com/andatoshiki/toshiki-notebook"}]},"features":[{"icon":"🕒","title":"Tempus Fugit","details":"Tempus Fugit, which means \\"time flies\\" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential."},{"icon":"☕","title":"Carpe Diem","details":"Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning."},{"icon":"💡","title":"Epiphania","details":"An epiphany, derived from the Latin word \\"epiphania,\\" is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas."}]},"headers":[],"relativePath":"index.md","filePath":"index.md","lastUpdated":1684859551000}`),i={name:"index.md"};function n(o,s,r,l,d,h){return a(),t("div")}const p=e(i,[["render",n]]);export{c as __pageData,p as default};
diff --git a/assets/javascript_notes_1_1-1.md.10b5b6ea.js b/assets/javascript_notes_1_1-1.md.7c957dab.js
similarity index 99%
rename from assets/javascript_notes_1_1-1.md.10b5b6ea.js
rename to assets/javascript_notes_1_1-1.md.7c957dab.js
index 33032fcc..0980884b 100644
--- a/assets/javascript_notes_1_1-1.md.10b5b6ea.js
+++ b/assets/javascript_notes_1_1-1.md.7c957dab.js
@@ -1,4 +1,4 @@
-import{_ as t,c as l,z as s,a as e,E as o,R as a,o as p,G as i}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"1-1: Values","description":"Chapter 1-1 notes on values","frontmatter":{"title":"1-1: Values","description":"Chapter 1-1 notes on values"},"headers":[],"relativePath":"javascript/notes/1/1-1.md","filePath":"javascript/notes/1/1-1.md","lastUpdated":1684858945000}'),r={name:"javascript/notes/1/1-1.md"},c=s("h1",{id:"_1-1-values",tabindex:"-1"},[e("1-1: Values "),s("a",{class:"header-anchor",href:"#_1-1-values","aria-label":'Permalink to "1-1: Values"'},"​")],-1),d={id:"_1-1-1-numbers",tabindex:"-1"},h=s("a",{class:"header-anchor",href:"#_1-1-1-numbers","aria-label":'Permalink to "1-1-1: Numbers "'},"​",-1),u=a(`
Values of the number type are numeric values, normally written as follows,
js
144
144
  • Put this into a program, it will cause the number 144 to come into existence inside the computer, with the following bash script, 144 might looks like this in bits,
    js
    0100000001100010000000000000000000000000000000000000000000000000
    0100000001100010000000000000000000000000000000000000000000000000
  • With the following bash script, 144 will be converted into binary values in integer form,
    sh
    ip1=144 # defines a variable named ip1, with the value of 10
+import{_ as t,c as l,z as s,a as e,E as o,R as a,o as p,G as i}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"1-1: Values","description":"Chapter 1-1 notes on values","frontmatter":{"title":"1-1: Values","description":"Chapter 1-1 notes on values"},"headers":[],"relativePath":"javascript/notes/1/1-1.md","filePath":"javascript/notes/1/1-1.md","lastUpdated":1684859551000}'),r={name:"javascript/notes/1/1-1.md"},c=s("h1",{id:"_1-1-values",tabindex:"-1"},[e("1-1: Values "),s("a",{class:"header-anchor",href:"#_1-1-values","aria-label":'Permalink to "1-1: Values"'},"​")],-1),d={id:"_1-1-1-numbers",tabindex:"-1"},h=s("a",{class:"header-anchor",href:"#_1-1-1-numbers","aria-label":'Permalink to "1-1-1: Numbers "'},"​",-1),u=a(`
    Values of the number type are numeric values, normally written as follows,
    js
    144
    144
    • Put this into a program, it will cause the number 144 to come into existence inside the computer, with the following bash script, 144 might looks like this in bits,
      js
      0100000001100010000000000000000000000000000000000000000000000000
      0100000001100010000000000000000000000000000000000000000000000000
    • With the following bash script, 144 will be converted into binary values in integer form,
      sh
      ip1=144 # defines a variable named ip1, with the value of 10
 echo "obase=2;$ip1" | bc # convert the value to binary via bc
      ip1=144 # defines a variable named ip1, with the value of 10
 echo "obase=2;$ip1" | bc # convert the value to binary via bc
      • The variable ip1 is assigned the value 144.
      • The echo command prints out the string "obase=2;144", where "obase=2;" is an argument for bc that tells it to output the result in base 2 (binary), and "10" is the decimal number we want to convert.
      • The output of the echo command is piped (using the | character) to bc, which takes the input "obase=2;144" and interprets it as a command to convert the number 10 to binary.
      • Finally, the binary equivalent of 144, which is "10010000", is printed to the terminal.
    TIP
    But the standard describes JavaScript numbers as 64-bit floating-point values, which indicates fractions and exponents are available.
    `,4),m=s("p",null,[e("Not all whole numbers "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"19")])]),s("annotation",{encoding:"application/x-tex"},"10^{19}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"19")])])])])])])])])])])]),e(" fit in a JavaScript number. There are also negative numbers, so one of the bits has to be used to store the sign of the number.")],-1),y=s("ul",null,[s("li",null,"11 bits are used to store the position of the decimal dot within the number."),s("li",null,[e("52 bits, any whole number less than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(", which is more than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"15")])]),s("annotation",{encoding:"application/x-tex"},"10^{15}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"15")])])])])])])])])])])]),e(" will safely fit in a JavaScript number, numbers we are using stay well below that.")])],-1),g=a('
    Fractional numbers are written by using a dot,
    js
    9.81
    9.81
    For extreme huge or tiny numbers, we can also use "scientific" notion by adding an e, followed by the exponent of the number,
    js
    2.998e8
    2.998e8
    ',4),b=s("p",null,[e("This indicates "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.998"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"8")]),s("mo",null,"="),s("mn",null,"29980000")]),s("annotation",{encoding:"application/x-tex"},"2.998 \\times 10^8 = 29980000")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2.998"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"8")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"29980000")])])]),e(".")],-1),v=s("p",null,[e("Calculations with whole numbers (referred as "),s("em",null,"integers"),e(") that fits into 52 bits are guaranteed to be always precise, but calculation with fraction numbers are generally not.")],-1),C=s("ul",null,[s("li",null,[e("Such as "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),e(" cannot be precisely expressed by a finite amount of decimal digits, "),s("strong",null,"thus many numbers lose some precision when only 64 bits are available to store them.")])],-1),w=s("details",{class:"details custom-block"},[s("summary",null,"1-1-1: Numbers review"),s("ul",null,[s("li",null,[e("64 bits: A 64-bit number is a binary sequence of 64 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"64")])]),s("annotation",{encoding:"application/x-tex"},"2^{64}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"64")])])])])])])])])])])]),e(" (18,446,744,073,709,551,616) different values. This is commonly used for representing memory addresses and integers with large values in computer systems.")]),s("li",null,[e("11 bits: An 11-bit number is a binary sequence of 11 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"11")])]),s("annotation",{encoding:"application/x-tex"},"2^{11}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"11")])])])])])])])])])])]),e(" (2,048) different values. This is often used in computer systems for encoding small integers, such as color values in images.")]),s("li",null,[e("52 bits: A 52-bit number is a binary sequence of 52 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(" (4,503,599,627,370,496) different values. This is often used for representing the significand or mantissa portion of a floating-point number in computer systems.")])])],-1),k=a(`
    1-1-2: Arithmetic ​
    The main thing to do with numbers is arithmetic. Arithmetic operations such as addition or multiplication take two number values and produce a new number from them, the following is an example of calculation in JavaScript.
    js
    100 + 4 * 11
    100 + 4 * 11
    • The + and * symbols are called operators.
      • + apparently stands for addition while * stands for multiplication.
    • This expression indicates the multiplication takes place first then the adding of 100 comes next as a regard to PEMDAS rules (Order of Operations).
    But we can still overwrite the steps of operation using a parenthesis around the addition,
    js
    ;(100 + 4) * 11
    ;(100 + 4) * 11
    For subtraction, there is the - operator, and division can be done with /. When operators appear together without parentheses, they are applied is determined by the precedence of the operators.
    • When multiple operators with the same precedence appear next to each other (as 1-2+1), they are applied left to right (refers to PEMDAS).
    • Precedence: Priorities.
    WARNING
    When is doubt, don't care about precedence, just add parentheses for order of operation.
    There is one more arithmetic operator, which is the % percent sign used to represent the modulo operation. The term "X modulo Y" is defined as the remainder of dividing X by Y.
    • For example, 314 % 100 is 14, 10 % 3 is 1, and 144 % 12 is 0 because there are no remainders.
    • Modulo's precedence is the same as that of multiplication and division.
    1-2: Arithmetic review
    • All of the operators in JavaScript follows precedence rules, or referred as PEMDAS in conventional math. 
      • + and - has the same lowest precedence.
      • * , / and % has the same medium precedence.
      • () has the highest precedence, followed on with brackets.
    • The symbol % refers to "modulo", in mathematical reading, we say "X modulo Y" is the remainder of X over Y.
    1-1-3: Strings ​
    The next data type is the string. Its use is not as evident from its name as with numbers, it also fulfills a very basic role.
    • Strings are used to represent text.
    • Strings are usually written by enclosing the contents with quotes.
    js
    'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'
 'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'
    'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'
diff --git a/assets/javascript_notes_1_1-1.md.10b5b6ea.lean.js b/assets/javascript_notes_1_1-1.md.7c957dab.lean.js
similarity index 99%
rename from assets/javascript_notes_1_1-1.md.10b5b6ea.lean.js
rename to assets/javascript_notes_1_1-1.md.7c957dab.lean.js
index d5633144..3d301f29 100644
--- a/assets/javascript_notes_1_1-1.md.10b5b6ea.lean.js
+++ b/assets/javascript_notes_1_1-1.md.7c957dab.lean.js
@@ -1 +1 @@
-import{_ as t,c as l,z as s,a as e,E as o,R as a,o as p,G as i}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"1-1: Values","description":"Chapter 1-1 notes on values","frontmatter":{"title":"1-1: Values","description":"Chapter 1-1 notes on values"},"headers":[],"relativePath":"javascript/notes/1/1-1.md","filePath":"javascript/notes/1/1-1.md","lastUpdated":1684858945000}'),r={name:"javascript/notes/1/1-1.md"},c=s("h1",{id:"_1-1-values",tabindex:"-1"},[e("1-1: Values "),s("a",{class:"header-anchor",href:"#_1-1-values","aria-label":'Permalink to "1-1: Values"'},"​")],-1),d={id:"_1-1-1-numbers",tabindex:"-1"},h=s("a",{class:"header-anchor",href:"#_1-1-1-numbers","aria-label":'Permalink to "1-1-1: Numbers "'},"​",-1),u=a("",4),m=s("p",null,[e("Not all whole numbers "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"19")])]),s("annotation",{encoding:"application/x-tex"},"10^{19}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"19")])])])])])])])])])])]),e(" fit in a JavaScript number. There are also negative numbers, so one of the bits has to be used to store the sign of the number.")],-1),y=s("ul",null,[s("li",null,"11 bits are used to store the position of the decimal dot within the number."),s("li",null,[e("52 bits, any whole number less than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(", which is more than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"15")])]),s("annotation",{encoding:"application/x-tex"},"10^{15}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"15")])])])])])])])])])])]),e(" will safely fit in a JavaScript number, numbers we are using stay well below that.")])],-1),g=a("",4),b=s("p",null,[e("This indicates "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.998"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"8")]),s("mo",null,"="),s("mn",null,"29980000")]),s("annotation",{encoding:"application/x-tex"},"2.998 \\times 10^8 = 29980000")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2.998"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"8")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"29980000")])])]),e(".")],-1),v=s("p",null,[e("Calculations with whole numbers (referred as "),s("em",null,"integers"),e(") that fits into 52 bits are guaranteed to be always precise, but calculation with fraction numbers are generally not.")],-1),C=s("ul",null,[s("li",null,[e("Such as "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),e(" cannot be precisely expressed by a finite amount of decimal digits, "),s("strong",null,"thus many numbers lose some precision when only 64 bits are available to store them.")])],-1),w=s("details",{class:"details custom-block"},[s("summary",null,"1-1-1: Numbers review"),s("ul",null,[s("li",null,[e("64 bits: A 64-bit number is a binary sequence of 64 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"64")])]),s("annotation",{encoding:"application/x-tex"},"2^{64}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"64")])])])])])])])])])])]),e(" (18,446,744,073,709,551,616) different values. This is commonly used for representing memory addresses and integers with large values in computer systems.")]),s("li",null,[e("11 bits: An 11-bit number is a binary sequence of 11 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"11")])]),s("annotation",{encoding:"application/x-tex"},"2^{11}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"11")])])])])])])])])])])]),e(" (2,048) different values. This is often used in computer systems for encoding small integers, such as color values in images.")]),s("li",null,[e("52 bits: A 52-bit number is a binary sequence of 52 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(" (4,503,599,627,370,496) different values. This is often used for representing the significand or mantissa portion of a floating-point number in computer systems.")])])],-1),k=a("",37);function A(D,F,B,f,_,q){const n=i("Badge");return p(),l("div",null,[c,s("h2",d,[e("1-1-1: Numbers "),o(n,{type:"danger",text:"must know"}),e(),h]),u,m,y,g,b,v,C,w,k])}const E=t(r,[["render",A]]);export{T as __pageData,E as default};
+import{_ as t,c as l,z as s,a as e,E as o,R as a,o as p,G as i}from"./chunks/framework.a1bac067.js";const T=JSON.parse('{"title":"1-1: Values","description":"Chapter 1-1 notes on values","frontmatter":{"title":"1-1: Values","description":"Chapter 1-1 notes on values"},"headers":[],"relativePath":"javascript/notes/1/1-1.md","filePath":"javascript/notes/1/1-1.md","lastUpdated":1684859551000}'),r={name:"javascript/notes/1/1-1.md"},c=s("h1",{id:"_1-1-values",tabindex:"-1"},[e("1-1: Values "),s("a",{class:"header-anchor",href:"#_1-1-values","aria-label":'Permalink to "1-1: Values"'},"​")],-1),d={id:"_1-1-1-numbers",tabindex:"-1"},h=s("a",{class:"header-anchor",href:"#_1-1-1-numbers","aria-label":'Permalink to "1-1-1: Numbers "'},"​",-1),u=a("",4),m=s("p",null,[e("Not all whole numbers "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"19")])]),s("annotation",{encoding:"application/x-tex"},"10^{19}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"19")])])])])])])])])])])]),e(" fit in a JavaScript number. There are also negative numbers, so one of the bits has to be used to store the sign of the number.")],-1),y=s("ul",null,[s("li",null,"11 bits are used to store the position of the decimal dot within the number."),s("li",null,[e("52 bits, any whole number less than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(", which is more than "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"15")])]),s("annotation",{encoding:"application/x-tex"},"10^{15}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"15")])])])])])])])])])])]),e(" will safely fit in a JavaScript number, numbers we are using stay well below that.")])],-1),g=a("",4),b=s("p",null,[e("This indicates "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mn",null,"2.998"),s("mo",null,"×"),s("mn",null,"1"),s("msup",null,[s("mn",null,"0"),s("mn",null,"8")]),s("mo",null,"="),s("mn",null,"29980000")]),s("annotation",{encoding:"application/x-tex"},"2.998 \\times 10^8 = 29980000")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7278em","vertical-align":"-0.0833em"}}),s("span",{class:"mord"},"2.998"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},"×"),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},"1"),s("span",{class:"mord"},[s("span",{class:"mord"},"0"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},"8")])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.6444em"}}),s("span",{class:"mord"},"29980000")])])]),e(".")],-1),v=s("p",null,[e("Calculations with whole numbers (referred as "),s("em",null,"integers"),e(") that fits into 52 bits are guaranteed to be always precise, but calculation with fraction numbers are generally not.")],-1),C=s("ul",null,[s("li",null,[e("Such as "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("mi",null,"π")]),s("annotation",{encoding:"application/x-tex"},"\\pi")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal",style:{"margin-right":"0.03588em"}},"π")])])]),e(" cannot be precisely expressed by a finite amount of decimal digits, "),s("strong",null,"thus many numbers lose some precision when only 64 bits are available to store them.")])],-1),w=s("details",{class:"details custom-block"},[s("summary",null,"1-1-1: Numbers review"),s("ul",null,[s("li",null,[e("64 bits: A 64-bit number is a binary sequence of 64 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"64")])]),s("annotation",{encoding:"application/x-tex"},"2^{64}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"64")])])])])])])])])])])]),e(" (18,446,744,073,709,551,616) different values. This is commonly used for representing memory addresses and integers with large values in computer systems.")]),s("li",null,[e("11 bits: An 11-bit number is a binary sequence of 11 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"11")])]),s("annotation",{encoding:"application/x-tex"},"2^{11}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"11")])])])])])])])])])])]),e(" (2,048) different values. This is often used in computer systems for encoding small integers, such as color values in images.")]),s("li",null,[e("52 bits: A 52-bit number is a binary sequence of 52 bits, which can represent "),s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("semantics",null,[s("mrow",null,[s("msup",null,[s("mn",null,"2"),s("mn",null,"52")])]),s("annotation",{encoding:"application/x-tex"},"2^{52}")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8141em"}}),s("span",{class:"mord"},[s("span",{class:"mord"},"2"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8141em"}},[s("span",{style:{top:"-3.063em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mtight"},"52")])])])])])])])])])])]),e(" (4,503,599,627,370,496) different values. This is often used for representing the significand or mantissa portion of a floating-point number in computer systems.")])])],-1),k=a("",37);function A(D,F,B,f,_,q){const n=i("Badge");return p(),l("div",null,[c,s("h2",d,[e("1-1-1: Numbers "),o(n,{type:"danger",text:"must know"}),e(),h]),u,m,y,g,b,v,C,w,k])}const E=t(r,[["render",A]]);export{T as __pageData,E as default};
diff --git a/assets/javascript_notes_1_1-2.md.5a4dd0dd.js b/assets/javascript_notes_1_1-2.md.46f1230f.js
similarity index 84%
rename from assets/javascript_notes_1_1-2.md.5a4dd0dd.js
rename to assets/javascript_notes_1_1-2.md.46f1230f.js
index 511a4e05..07673e4a 100644
--- a/assets/javascript_notes_1_1-2.md.5a4dd0dd.js
+++ b/assets/javascript_notes_1_1-2.md.46f1230f.js
@@ -1 +1 @@
-import{_ as t,c as e,o as a}from"./chunks/framework.a1bac067.js";const l=JSON.parse('{"title":"1-2","description":"","frontmatter":{"title":"1-2"},"headers":[],"relativePath":"javascript/notes/1/1-2.md","filePath":"javascript/notes/1/1-2.md","lastUpdated":1684858945000}'),s={name:"javascript/notes/1/1-2.md"};function r(o,c,n,p,i,d){return a(),e("div")}const m=t(s,[["render",r]]);export{l as __pageData,m as default};
+import{_ as t,c as e,o as a}from"./chunks/framework.a1bac067.js";const l=JSON.parse('{"title":"1-2","description":"","frontmatter":{"title":"1-2"},"headers":[],"relativePath":"javascript/notes/1/1-2.md","filePath":"javascript/notes/1/1-2.md","lastUpdated":1684859551000}'),s={name:"javascript/notes/1/1-2.md"};function r(o,c,n,p,i,d){return a(),e("div")}const m=t(s,[["render",r]]);export{l as __pageData,m as default};
diff --git a/assets/javascript_notes_1_1-2.md.5a4dd0dd.lean.js b/assets/javascript_notes_1_1-2.md.46f1230f.lean.js
similarity index 84%
rename from assets/javascript_notes_1_1-2.md.5a4dd0dd.lean.js
rename to assets/javascript_notes_1_1-2.md.46f1230f.lean.js
index 511a4e05..07673e4a 100644
--- a/assets/javascript_notes_1_1-2.md.5a4dd0dd.lean.js
+++ b/assets/javascript_notes_1_1-2.md.46f1230f.lean.js
@@ -1 +1 @@
-import{_ as t,c as e,o as a}from"./chunks/framework.a1bac067.js";const l=JSON.parse('{"title":"1-2","description":"","frontmatter":{"title":"1-2"},"headers":[],"relativePath":"javascript/notes/1/1-2.md","filePath":"javascript/notes/1/1-2.md","lastUpdated":1684858945000}'),s={name:"javascript/notes/1/1-2.md"};function r(o,c,n,p,i,d){return a(),e("div")}const m=t(s,[["render",r]]);export{l as __pageData,m as default};
+import{_ as t,c as e,o as a}from"./chunks/framework.a1bac067.js";const l=JSON.parse('{"title":"1-2","description":"","frontmatter":{"title":"1-2"},"headers":[],"relativePath":"javascript/notes/1/1-2.md","filePath":"javascript/notes/1/1-2.md","lastUpdated":1684859551000}'),s={name:"javascript/notes/1/1-2.md"};function r(o,c,n,p,i,d){return a(),e("div")}const m=t(s,[["render",r]]);export{l as __pageData,m as default};
diff --git a/assets/roadmap.md.7ba79349.js b/assets/roadmap.md.a42ebae3.js
similarity index 98%
rename from assets/roadmap.md.7ba79349.js
rename to assets/roadmap.md.a42ebae3.js
index 7ccece65..611c9dcd 100644
--- a/assets/roadmap.md.7ba79349.js
+++ b/assets/roadmap.md.a42ebae3.js
@@ -1 +1 @@
-import{_ as t,c as e,o as s,R as i}from"./chunks/framework.a1bac067.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[],"relativePath":"roadmap.md","filePath":"roadmap.md","lastUpdated":1684858945000}'),a={name:"roadmap.md"},l=i('
    Theme ​
    Homepage ​
    •  Finish three hero action blocks with keywords and its corresponded descriptions.
    •  Finish my learning platform section using SVGs of different online academies from simple icons.
      • Descriptions
      • Icon in SVG format
      • Platform name
      • Linkable SVG icon/button
        •  Khan academy
        •  GitHub
        •  Libretext
        •  OpenStax
        •  BCcampus
        •  Wikibooks
        •  Wikipedia
        •  Bookboon
        • [ ]
    Configs ​
    •  Header configs including different properties and meta configs.
    •  Complete navigation configuration. 
    •  Complete sidebar configs for different directories.
    •  Vitepress sidebar generation plugin for huge bundled clips article directory.
    Articles ​
    •  <head> tag documentation or guide by joshbuchea/HEAD.
    •  Write index page for every directory, no Chat GPT allowed.
    •  Chemistry review notes.
    •  JavaScript 0-100 notes per chapter. 
      • Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)
    •  Outliers novel clips
    •  Migrate clipped articles and archives from Notion to Vitepress.
    •  Fresh copy of "How To Ask Questions The Smart Way" by Eric Steven Raymond 
    ',9),c=[l];function o(r,n,d,h,k,p){return s(),e("div",null,c)}const u=t(a,[["render",o]]);export{m as __pageData,u as default};
+import{_ as t,c as e,o as s,R as i}from"./chunks/framework.a1bac067.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[],"relativePath":"roadmap.md","filePath":"roadmap.md","lastUpdated":1684859551000}'),a={name:"roadmap.md"},l=i('
    Theme ​
    Homepage ​
    •  Finish three hero action blocks with keywords and its corresponded descriptions.
    •  Finish my learning platform section using SVGs of different online academies from simple icons.
      • Descriptions
      • Icon in SVG format
      • Platform name
      • Linkable SVG icon/button
        •  Khan academy
        •  GitHub
        •  Libretext
        •  OpenStax
        •  BCcampus
        •  Wikibooks
        •  Wikipedia
        •  Bookboon
        • [ ]
    Configs ​
    •  Header configs including different properties and meta configs.
    •  Complete navigation configuration. 
    •  Complete sidebar configs for different directories.
    •  Vitepress sidebar generation plugin for huge bundled clips article directory.
    Articles ​
    •  <head> tag documentation or guide by joshbuchea/HEAD.
    •  Write index page for every directory, no Chat GPT allowed.
    •  Chemistry review notes.
    •  JavaScript 0-100 notes per chapter. 
      • Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)
    •  Outliers novel clips
    •  Migrate clipped articles and archives from Notion to Vitepress.
    •  Fresh copy of "How To Ask Questions The Smart Way" by Eric Steven Raymond 
    ',9),c=[l];function o(r,n,d,h,k,p){return s(),e("div",null,c)}const u=t(a,[["render",o]]);export{m as __pageData,u as default};
diff --git a/assets/roadmap.md.7ba79349.lean.js b/assets/roadmap.md.a42ebae3.lean.js
similarity index 83%
rename from assets/roadmap.md.7ba79349.lean.js
rename to assets/roadmap.md.a42ebae3.lean.js
index bf11722a..74f841e4 100644
--- a/assets/roadmap.md.7ba79349.lean.js
+++ b/assets/roadmap.md.a42ebae3.lean.js
@@ -1 +1 @@
-import{_ as t,c as e,o as s,R as i}from"./chunks/framework.a1bac067.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[],"relativePath":"roadmap.md","filePath":"roadmap.md","lastUpdated":1684858945000}'),a={name:"roadmap.md"},l=i("",9),c=[l];function o(r,n,d,h,k,p){return s(),e("div",null,c)}const u=t(a,[["render",o]]);export{m as __pageData,u as default};
+import{_ as t,c as e,o as s,R as i}from"./chunks/framework.a1bac067.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[],"relativePath":"roadmap.md","filePath":"roadmap.md","lastUpdated":1684859551000}'),a={name:"roadmap.md"},l=i("",9),c=[l];function o(r,n,d,h,k,p){return s(),e("div",null,c)}const u=t(a,[["render",o]]);export{m as __pageData,u as default};
diff --git a/assets/save_reading_index.md.78fc5e98.js b/assets/save_reading_index.md.983fe09d.js
similarity index 90%
rename from assets/save_reading_index.md.78fc5e98.js
rename to assets/save_reading_index.md.983fe09d.js
index 3ae884fd..258e1ee3 100644
--- a/assets/save_reading_index.md.78fc5e98.js
+++ b/assets/save_reading_index.md.983fe09d.js
@@ -1 +1 @@
-import{_ as a,c as t,o,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","filePath":"save/reading/index.md","lastUpdated":1684858945000}'),s={name:"save/reading/index.md"},d=e("h1",{id:"welcome-to-reading-notes",tabindex:"-1"},[n("Welcome to Reading Notes "),e("a",{class:"header-anchor",href:"#welcome-to-reading-notes","aria-label":'Permalink to "Welcome to Reading Notes"'},"​")],-1),r=[d];function i(c,l,m,_,p,h){return o(),t("div",null,r)}const x=a(s,[["render",i]]);export{g as __pageData,x as default};
+import{_ as a,c as t,o,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","filePath":"save/reading/index.md","lastUpdated":1684859551000}'),s={name:"save/reading/index.md"},d=e("h1",{id:"welcome-to-reading-notes",tabindex:"-1"},[n("Welcome to Reading Notes "),e("a",{class:"header-anchor",href:"#welcome-to-reading-notes","aria-label":'Permalink to "Welcome to Reading Notes"'},"​")],-1),r=[d];function i(c,l,m,_,p,h){return o(),t("div",null,r)}const x=a(s,[["render",i]]);export{g as __pageData,x as default};
diff --git a/assets/save_reading_index.md.78fc5e98.lean.js b/assets/save_reading_index.md.983fe09d.lean.js
similarity index 90%
rename from assets/save_reading_index.md.78fc5e98.lean.js
rename to assets/save_reading_index.md.983fe09d.lean.js
index 3ae884fd..258e1ee3 100644
--- a/assets/save_reading_index.md.78fc5e98.lean.js
+++ b/assets/save_reading_index.md.983fe09d.lean.js
@@ -1 +1 @@
-import{_ as a,c as t,o,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","filePath":"save/reading/index.md","lastUpdated":1684858945000}'),s={name:"save/reading/index.md"},d=e("h1",{id:"welcome-to-reading-notes",tabindex:"-1"},[n("Welcome to Reading Notes "),e("a",{class:"header-anchor",href:"#welcome-to-reading-notes","aria-label":'Permalink to "Welcome to Reading Notes"'},"​")],-1),r=[d];function i(c,l,m,_,p,h){return o(),t("div",null,r)}const x=a(s,[["render",i]]);export{g as __pageData,x as default};
+import{_ as a,c as t,o,z as e,a as n}from"./chunks/framework.a1bac067.js";const g=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","filePath":"save/reading/index.md","lastUpdated":1684859551000}'),s={name:"save/reading/index.md"},d=e("h1",{id:"welcome-to-reading-notes",tabindex:"-1"},[n("Welcome to Reading Notes "),e("a",{class:"header-anchor",href:"#welcome-to-reading-notes","aria-label":'Permalink to "Welcome to Reading Notes"'},"​")],-1),r=[d];function i(c,l,m,_,p,h){return o(),t("div",null,r)}const x=a(s,[["render",i]]);export{g as __pageData,x as default};
diff --git a/assets/save_reading_outliers_1.md.daf781a1.js b/assets/save_reading_outliers_1.md.e3cd2c6d.js
similarity index 99%
rename from assets/save_reading_outliers_1.md.daf781a1.js
rename to assets/save_reading_outliers_1.md.e3cd2c6d.js
index 13798a2a..a5acacad 100644
--- a/assets/save_reading_outliers_1.md.daf781a1.js
+++ b/assets/save_reading_outliers_1.md.e3cd2c6d.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells","frontmatter":{"author":"查尔斯","title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells"},"headers":[],"relativePath":"save/reading/outliers/1.md","filePath":"save/reading/outliers/1.md","lastUpdated":1684858945000}'),s={name:"save/reading/outliers/1.md"},r=o('
    Introduction & Chapter 1: The Roseto Mystery ​
    Summary: Introduction: The Roseto Mystery ​
    Section 1. ​
    In the late 1800s, immigrants from Roseto Valfortore, Italy, came to Pennsylvania to work in the slate quarry near the town of Bangor. The settlers established a new community, named after their old one: Roseto. They built schools, a park, small shops, and more than a dozen factories. They kept to themselves and did not interact much with the predominantly German and English populations of the towns around them.
    In the 1950s, a local doctor told Stewart Wolf, a medical researcher, that hardly anyone in Roseto under age sixty-five had heart disease. Wolf was curious. He did blood draws and EKG scans, and he analyzed physicians’ records and death certificates. Wolf was amazed that almost no one under fifty-five showed any signs of heart disease, which at the time was America’s leading cause of death in men under sixty-five. Wolf enlisted the help of John Bruhn, a sociologist. They found that there was no suicide, alcoholism, or drug addiction in Roseto: everyone seemed to be dying of old age. According to Gladwell, Roseto was an outlier.
    Section 2. ​
    Wolf was forced to rule out diet and lifestyle as explanations, because the Rosetans had poor eating habits, smoked heavily, and were obese. He ruled out genetics after determining that Rosetans’ relatives in other parts of the country were not in comparably good health. Wolf eventually decided that what made Roseto’s inhabitants so healthy was the town itself—its culture and social structure. Many homes contained three generations living together, and with a population of just two thousand people, there were twenty-two separate civic organizations.
    Wolf and Bruhn were met with a great deal of skepticism. They had to work hard to convince the medical establishment that the values of where we live and the people around us have a significant impact on who we are. Gladwell wants Outliers to do for our understanding of success what Wolf did for our understanding of health.
    Summary: Chapter One: The Matthew Effect ​
    Section 1. ​
    Gladwell describes a game between two teams in the Canadian Hockey League, the most competitive junior hockey league in the world. Starting at a very young age, players are regularly assessed for talent, so that the best can be separated out and prepared for the next level. The system is designed to be a meritocracy.
    Section 2. ​
    Gladwell wants to show that circumstances matter just as much for success as work ethic or intelligence. People often ask about the lifestyle, intelligence, or special talents of successful individuals, but not enough people ask when and where a person grew up. Gladwell likens people to trees: we all know that the tallest trees received the most sunlight, had nutrient-rich soil, and were lucky enough to avoid lumberjacks. He suggests that we study the environmental and social influences that make people successful.
    Section 3. ​
    In the 1980s, Canadian psychologist Roger Barnsley observed that on elite Canadian hockey teams, whether youth league or NHL, a disproportionate number of players have birthdays in the first three months of the year.
    Section 4. ​
    The explanation for the distribution of birthdays is simple: the cutoff date for an individual’s league assignment is January 1. A player born on January 2 will be 12 months older than one born at the end of December. At the age of nine or ten, this often translates into a huge difference in size, coordination, and maturity. By age ten, coaches regularly pick the oldest and largest potential players for the more competitive “rep squads,” where the players will receive better coaching, have more practice time, and play many more games. By thirteen or fourteen, all of the extra experience and coaching will have actually made the players better, and they will be selected for the more competitive league(s). The same thing happens in American youth baseball and European soccer, but with different cutoff dates.
    The phenomenon is also observable in education. Economists Kelly Bedard and Elizabeth Dhuey found that on math and science tests, the older children in fourth grade regularly score better than the youngest children. This can affect whether a child qualifies for a gifted program. When Bedard and Dhuey extended their analysis to four-year colleges, they found that the students who belonged to the relatively youngest group were underrepresented by more than 10 percent.
    Section 5. ​
    The top-level hockey players were given opportunities they neither deserved nor earned. Sociologist Robert Merton called this the “Matthew Effect,” after the verse in the Gospel of Matthew:
    “For unto everyone that hath shall be given, and he shall have abundance. But from that hath not shall be taken away even that which he hath.”
    The successful receive more resources: the best students receive the best teaching, the richest citizens get the biggest tax breaks, the largest kids get the best coaching. Sociologists call this “accumulative advantage.” Gladwell points out that cutoff dates exclude nearly half of the potential athletes from a population. Junior hockey teams in the Czech Republic show the same unbalanced distribution of birthdays as in Canada. If Canada or the Czech Republic separated each of their youth leagues into two separate leagues, one for kids born in the first half of the year, and one for kids born in the second half of the year, they would have twice as many athletes available for their national teams. Similarly, schools could separate kindergarten classes into students born in the first, middle, and last four months of the year. The benefits would justify the added administrative work, but, writes Gladwell, the scheme would run afoul of society’s need to believe that individual merit matters more than the rules society has created.
    Section 6. ​
    Gord Wasden, the father of one of the Canadian Hockey League boys from the first section, describes his son. Scott Wasden worked hard to get where he is, but he has also generally had the advantage of being big compared to others on his team. He was born on January 4.
    Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect ​
    Gladwell's thesis argues that that the idea of rugged, individual success is not accurate. Rather, the most successful person doesn’t thrive without some environmental and social influence plus a dose of good fortune.  After laying out the concepts of a meritocracy and suggesting that this is the way that hockey players advance in Canada, he immediately undercuts this idea. Gladwell uses the illustration of birthday distribution as an example of an invisible contributor to the success of hockey players. With a registration cutoff date of January 1, players born from January 2 through March 31 are older than teammates born later in the year. These older players are likely bigger, faster, more coordinated, and more mature. Since the older players are seen as more skilled, they get selected for more advanced leagues with more practice, more games, and higher-quality coaching. By the time these older players reach their early teens, the extra advantages gained from being judged better because of their birthday translate into being better players. Similarly, the age cutoff for school registrations often mistakes maturity for ability. As a result, the older children appear smarter, get put into advanced groups, and qualify for gifted programs. The birthdate bias carries through to college.
    Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that "success leads to more success." More simply: by being a little bit better, a hockey player will get opportunities that may result in the player becoming an outlier. Gladwell implies that the systems that determine success are not efficient. He asks readers to question the way we, as a society, think about success. He suggests that changing the systems in athletics and education would chip away at the myth of individual merit as the chief marker of success and help level the playing field. Gladwell's model for this phenomenon is hockey player Scott Wasden. By demonstrating that Scott had the passion, talent, and work ethic for the game, as well as the good luck to be born on January 4, Gladwell reasons that Wasden, like many others he will be profiling, benefit by being both good and lucky.
    Reference
    • “Outliers Introduction & Chapter 1 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section1/.
    ',28),n=[r];function i(h,l,d,c,u,m){return a(),t("div",null,n)}const p=e(s,[["render",i]]);export{y as __pageData,p as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells","frontmatter":{"author":"查尔斯","title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells"},"headers":[],"relativePath":"save/reading/outliers/1.md","filePath":"save/reading/outliers/1.md","lastUpdated":1684859551000}'),s={name:"save/reading/outliers/1.md"},r=o('
    Introduction & Chapter 1: The Roseto Mystery ​
    Summary: Introduction: The Roseto Mystery ​
    Section 1. ​
    In the late 1800s, immigrants from Roseto Valfortore, Italy, came to Pennsylvania to work in the slate quarry near the town of Bangor. The settlers established a new community, named after their old one: Roseto. They built schools, a park, small shops, and more than a dozen factories. They kept to themselves and did not interact much with the predominantly German and English populations of the towns around them.
    In the 1950s, a local doctor told Stewart Wolf, a medical researcher, that hardly anyone in Roseto under age sixty-five had heart disease. Wolf was curious. He did blood draws and EKG scans, and he analyzed physicians’ records and death certificates. Wolf was amazed that almost no one under fifty-five showed any signs of heart disease, which at the time was America’s leading cause of death in men under sixty-five. Wolf enlisted the help of John Bruhn, a sociologist. They found that there was no suicide, alcoholism, or drug addiction in Roseto: everyone seemed to be dying of old age. According to Gladwell, Roseto was an outlier.
    Section 2. ​
    Wolf was forced to rule out diet and lifestyle as explanations, because the Rosetans had poor eating habits, smoked heavily, and were obese. He ruled out genetics after determining that Rosetans’ relatives in other parts of the country were not in comparably good health. Wolf eventually decided that what made Roseto’s inhabitants so healthy was the town itself—its culture and social structure. Many homes contained three generations living together, and with a population of just two thousand people, there were twenty-two separate civic organizations.
    Wolf and Bruhn were met with a great deal of skepticism. They had to work hard to convince the medical establishment that the values of where we live and the people around us have a significant impact on who we are. Gladwell wants Outliers to do for our understanding of success what Wolf did for our understanding of health.
    Summary: Chapter One: The Matthew Effect ​
    Section 1. ​
    Gladwell describes a game between two teams in the Canadian Hockey League, the most competitive junior hockey league in the world. Starting at a very young age, players are regularly assessed for talent, so that the best can be separated out and prepared for the next level. The system is designed to be a meritocracy.
    Section 2. ​
    Gladwell wants to show that circumstances matter just as much for success as work ethic or intelligence. People often ask about the lifestyle, intelligence, or special talents of successful individuals, but not enough people ask when and where a person grew up. Gladwell likens people to trees: we all know that the tallest trees received the most sunlight, had nutrient-rich soil, and were lucky enough to avoid lumberjacks. He suggests that we study the environmental and social influences that make people successful.
    Section 3. ​
    In the 1980s, Canadian psychologist Roger Barnsley observed that on elite Canadian hockey teams, whether youth league or NHL, a disproportionate number of players have birthdays in the first three months of the year.
    Section 4. ​
    The explanation for the distribution of birthdays is simple: the cutoff date for an individual’s league assignment is January 1. A player born on January 2 will be 12 months older than one born at the end of December. At the age of nine or ten, this often translates into a huge difference in size, coordination, and maturity. By age ten, coaches regularly pick the oldest and largest potential players for the more competitive “rep squads,” where the players will receive better coaching, have more practice time, and play many more games. By thirteen or fourteen, all of the extra experience and coaching will have actually made the players better, and they will be selected for the more competitive league(s). The same thing happens in American youth baseball and European soccer, but with different cutoff dates.
    The phenomenon is also observable in education. Economists Kelly Bedard and Elizabeth Dhuey found that on math and science tests, the older children in fourth grade regularly score better than the youngest children. This can affect whether a child qualifies for a gifted program. When Bedard and Dhuey extended their analysis to four-year colleges, they found that the students who belonged to the relatively youngest group were underrepresented by more than 10 percent.
    Section 5. ​
    The top-level hockey players were given opportunities they neither deserved nor earned. Sociologist Robert Merton called this the “Matthew Effect,” after the verse in the Gospel of Matthew:
    “For unto everyone that hath shall be given, and he shall have abundance. But from that hath not shall be taken away even that which he hath.”
    The successful receive more resources: the best students receive the best teaching, the richest citizens get the biggest tax breaks, the largest kids get the best coaching. Sociologists call this “accumulative advantage.” Gladwell points out that cutoff dates exclude nearly half of the potential athletes from a population. Junior hockey teams in the Czech Republic show the same unbalanced distribution of birthdays as in Canada. If Canada or the Czech Republic separated each of their youth leagues into two separate leagues, one for kids born in the first half of the year, and one for kids born in the second half of the year, they would have twice as many athletes available for their national teams. Similarly, schools could separate kindergarten classes into students born in the first, middle, and last four months of the year. The benefits would justify the added administrative work, but, writes Gladwell, the scheme would run afoul of society’s need to believe that individual merit matters more than the rules society has created.
    Section 6. ​
    Gord Wasden, the father of one of the Canadian Hockey League boys from the first section, describes his son. Scott Wasden worked hard to get where he is, but he has also generally had the advantage of being big compared to others on his team. He was born on January 4.
    Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect ​
    Gladwell's thesis argues that that the idea of rugged, individual success is not accurate. Rather, the most successful person doesn’t thrive without some environmental and social influence plus a dose of good fortune.  After laying out the concepts of a meritocracy and suggesting that this is the way that hockey players advance in Canada, he immediately undercuts this idea. Gladwell uses the illustration of birthday distribution as an example of an invisible contributor to the success of hockey players. With a registration cutoff date of January 1, players born from January 2 through March 31 are older than teammates born later in the year. These older players are likely bigger, faster, more coordinated, and more mature. Since the older players are seen as more skilled, they get selected for more advanced leagues with more practice, more games, and higher-quality coaching. By the time these older players reach their early teens, the extra advantages gained from being judged better because of their birthday translate into being better players. Similarly, the age cutoff for school registrations often mistakes maturity for ability. As a result, the older children appear smarter, get put into advanced groups, and qualify for gifted programs. The birthdate bias carries through to college.
    Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that "success leads to more success." More simply: by being a little bit better, a hockey player will get opportunities that may result in the player becoming an outlier. Gladwell implies that the systems that determine success are not efficient. He asks readers to question the way we, as a society, think about success. He suggests that changing the systems in athletics and education would chip away at the myth of individual merit as the chief marker of success and help level the playing field. Gladwell's model for this phenomenon is hockey player Scott Wasden. By demonstrating that Scott had the passion, talent, and work ethic for the game, as well as the good luck to be born on January 4, Gladwell reasons that Wasden, like many others he will be profiling, benefit by being both good and lucky.
    Reference
    • “Outliers Introduction & Chapter 1 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section1/.
    ',28),n=[r];function i(h,l,d,c,u,m){return a(),t("div",null,n)}const p=e(s,[["render",i]]);export{y as __pageData,p as default};
diff --git a/assets/save_reading_outliers_1.md.daf781a1.lean.js b/assets/save_reading_outliers_1.md.e3cd2c6d.lean.js
similarity index 91%
rename from assets/save_reading_outliers_1.md.daf781a1.lean.js
rename to assets/save_reading_outliers_1.md.e3cd2c6d.lean.js
index b57b9df3..05080380 100644
--- a/assets/save_reading_outliers_1.md.daf781a1.lean.js
+++ b/assets/save_reading_outliers_1.md.e3cd2c6d.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells","frontmatter":{"author":"查尔斯","title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells"},"headers":[],"relativePath":"save/reading/outliers/1.md","filePath":"save/reading/outliers/1.md","lastUpdated":1684858945000}'),s={name:"save/reading/outliers/1.md"},r=o("",28),n=[r];function i(h,l,d,c,u,m){return a(),t("div",null,n)}const p=e(s,[["render",i]]);export{y as __pageData,p as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const y=JSON.parse('{"title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells","frontmatter":{"author":"查尔斯","title":"Introduction & Chapter 1: The Roseto Mystery","description":"Chapter 1 summary clips from Sparknotes of novel Outliers by Malcom Gladwells"},"headers":[],"relativePath":"save/reading/outliers/1.md","filePath":"save/reading/outliers/1.md","lastUpdated":1684859551000}'),s={name:"save/reading/outliers/1.md"},r=o("",28),n=[r];function i(h,l,d,c,u,m){return a(),t("div",null,n)}const p=e(s,[["render",i]]);export{y as __pageData,p as default};
diff --git a/assets/save_reading_outliers_2.md.8febfe5f.js b/assets/save_reading_outliers_2.md.ad1783d7.js
similarity index 99%
rename from assets/save_reading_outliers_2.md.8febfe5f.js
rename to assets/save_reading_outliers_2.md.ad1783d7.js
index 8531d536..ed198c21 100644
--- a/assets/save_reading_outliers_2.md.8febfe5f.js
+++ b/assets/save_reading_outliers_2.md.ad1783d7.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes","frontmatter":{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/2.md","filePath":"save/reading/outliers/2.md","lastUpdated":1684858945000}'),i={name:"save/reading/outliers/2.md"},n=o('
    Chapter Two: The 10,000-Hour Rule ​
    Summary: Chapter Two: The 10,000-Hour Rule ​
    Section 1. ​
    Bill Joy, a co-founder of Sun Microsystems, is one of the most influential computer programmers of all time. Joy attended the University of Michigan, thinking to become a biologist or mathematician, but he became obsessed with the Computer Center that opened during his freshman year. In 1975, he enrolled in graduate school at UC Berkeley. At both schools, Bill Joy spent much of his free time programming. After graduating from Berkeley, he founded Sun Microsystems, eventually rewriting the Java programming language. He would appear to be a good example of success based on individual merit.
    Section 2. ​
    If achievement is the combination of talent and preparation, how much does innate talent affect one’s success? In the 1990s, a study at Berlin’s elite Academy of Music determined that, regardless of instrument, all of the world-class students had practiced more than 10,000 hours. They did not find anyone at the top level who spent less time, nor did they find anyone who spent that amount of time who was not successful. Neurologist Daniel Levitin states that, across many fields, world-class expertise is never accomplished in less than 10,000 hours. The law holds even for a prodigy like Mozart, who started composing at age six: he did not compose his first masterwork until he was twenty-one. The youth athletes from the previous chapter who made it to the professional level put in 10,000 hours of practice. And note: individuals cannot normally put in that much practice without help. Most people will need supportive parents, financial stability, and probably a special program—like an all-star youth hockey team.
    Section 3. ​
    Bill Joy was certainly intelligent, having achieved a perfect score on the math portion of his Scholastic Aptitude Test. His opportunities, however, were extraordinary. Bill Joy happened to choose a college that had a time-sharing computer available, something that was exceedingly rare during the early 1970s. While most students had to pay for computer time, Bill Joy exploited a bug in the college’s system to gain unlimited access to the Computer Center, which was walking distance from where he lived. He was able to spend almost all of his free time programming, and when he went to Berkeley, he had a terminal at home, where he could program until he fell asleep at his keyboard. Bill Joy was able to accumulate 10,000 hours while in college.
    Section 4. ​
    The Beatles are another example of the 10,000-hours principle. In 1960, when they were still a high school rock band, they were given the chance to travel to Hamburg, Germany and play in various strip clubs. The Beatles were contracted to play for hours every night, seven nights a week, and thereby were forced to develop as musicians. After a year and a half, they had performed 270 nights. By 1964, when they first found commercial success, they had performed an estimated twelve hundred times—more than most band bands do in an entire career.
    Section 5. ​
    Born in Seattle to affluent parents, Bill Gates was sent to a private school for seventh grade. The school’s Mother’s Club funded the purchase of a time-sharing terminal connected to a mainframe computer in Seattle. As an eighth-grader, Gates was able to start programming. When the Mother’s Club could no longer pay for computing time, Gates was able to continue programming at Information Sciences, Inc. (ISI) through a connection he made with one of the school parents. After that, one of ISI’s founders recommended Gates for a programming project at the Bonneville Power station. Gates was able to leave high school during his senior year to program for the Bonneville Power station as an independent study project. By the time Bill Gates dropped out of Harvard, he was well past 10,000 hours of experience.
    Section 6. ​
    In a list of the seventy-five richest people in human history, going as far back as ancient Egypt and converting all wealth estimates into modern U.S. dollars, fourteen names are Americans born within nine years of one another (including John D. Rockefeller and J.P. Morgan). What accounts for the fact that a list that covers thousands of years should have 20 percent of its members so close together? The fourteen Americans happened to live during the 1860s and 1870s, when the American economy was undergoing a huge transformation: the expansion of the railroads and the creation of Wall Street. Gladwell argues that the fourteen names are so close in birth date because they were all the perfect age to seize the opportunities available.
    Bill Joy and Bill Gates were also born at just the right time. In 1975, Popular Electronics ran a cover story about the Altair 8800, a $397 microcomputer kit. Someone born before 1954 would likely already have a job with IBM, working on mainframes. Someone born after 1956 would likely still be in high school. Most of the biggest pioneers in the modern computer industry, including Bill Gates, Bill Joy, Steve Jobs, and Eric Schmidt, were born in 1954 or 1955, as were Joy’s three fellow founders at Sun.
    Analysis: Chapter Two: The 10,000-Hour Rule ​
    Gladwell introduces his controversial and widely disputed 10,000 hours rule. Simply put, the rule suggests that to master a skill, an individual needs to practice it for 10,000 hours. Gladwell argues that, like hockey, success in computing requires both skill and luck, rather than pure, simple, natural ability. Gladwell's example is situated at the University of Michigan, specifically the new computer center, in 1971. The story of Bill Joy undermines the idea of success as an inexplicable, miraculous accomplishment. Though Gladwell agrees Joy is bright and talented, Joy also lucks into several happy accidents that allow for increased access. These strokes of luck give Joy the opportunity to become a computer expert. Gladwell's use of a psychological study conducted at the Academy of Music in Berlin, Germany, in the early 1990s forms the crux of his argument in this chapter. However, critics have suggested that this study was flawed and that there is no incontrovertible evidence to suggest that doing anything for 10,000 hours guarantees mastery. Rather, practice is, as Gladwell points out elsewhere throughout the text, just one of the factors in success. However, he also correlates the familial and economic advantages necessary to reach 10,000 of practice time, reinforcing the larger thesis of the book.
    Gladwell argues that the Beatles and Bill Gates are examples of his notion of time spent directly relating to mastery. While detailing how the Beatles reached the 10,000-hour threshold for mastery, Gladwell also highlights the lucky breaks they had in achieving success, like the connection that landed the band in Hamburg’s music scene. Additionally, while Gates has become something of an American folk hero, Gladwell argues that Gates did not reach his great achievements completely on his own. His wealthy family, his lucky interaction with a computer in 1968, and his college experience allowed him far greater access to opportunity. Without this combination of factors, along with his intelligence and drive, Gates would likely not have become a household name. Again, Gladwell reinforces the idea that success comes not only from being good, but from having no small amount of luck, whether that be time, money, or, in an echo of the hockey player example, birthdate.
    Gladwell claims that one important factor of success is the year in which one is born. In terms of wealth, there are 14 Americans, born in a nine-year period of the 1800s, who are among the richest people in history. This suggests that there was some connection to when they were born and the opportunities available to them in terms of innovation and invention, as well as the position they were in to take advantage of the opportunities. Similarly, men born in the same era of Bill Gates were coming of age at the bleeding edge of technology. Proximity to social and technological change, as well as the wealth and connections to take advantage of it, put these men in an advantageous position. Being in the right place at the right time, Gladwell argues, has at least as much to do with success as one's inborn abilities.
    Reference
    • “Outliers Chapter 2 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section2/. Accessed 27 Feb. 2023.
    ',20),r=[n];function s(h,l,c,d,u,m){return a(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes","frontmatter":{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/2.md","filePath":"save/reading/outliers/2.md","lastUpdated":1684859551000}'),i={name:"save/reading/outliers/2.md"},n=o('
    Chapter Two: The 10,000-Hour Rule ​
    Summary: Chapter Two: The 10,000-Hour Rule ​
    Section 1. ​
    Bill Joy, a co-founder of Sun Microsystems, is one of the most influential computer programmers of all time. Joy attended the University of Michigan, thinking to become a biologist or mathematician, but he became obsessed with the Computer Center that opened during his freshman year. In 1975, he enrolled in graduate school at UC Berkeley. At both schools, Bill Joy spent much of his free time programming. After graduating from Berkeley, he founded Sun Microsystems, eventually rewriting the Java programming language. He would appear to be a good example of success based on individual merit.
    Section 2. ​
    If achievement is the combination of talent and preparation, how much does innate talent affect one’s success? In the 1990s, a study at Berlin’s elite Academy of Music determined that, regardless of instrument, all of the world-class students had practiced more than 10,000 hours. They did not find anyone at the top level who spent less time, nor did they find anyone who spent that amount of time who was not successful. Neurologist Daniel Levitin states that, across many fields, world-class expertise is never accomplished in less than 10,000 hours. The law holds even for a prodigy like Mozart, who started composing at age six: he did not compose his first masterwork until he was twenty-one. The youth athletes from the previous chapter who made it to the professional level put in 10,000 hours of practice. And note: individuals cannot normally put in that much practice without help. Most people will need supportive parents, financial stability, and probably a special program—like an all-star youth hockey team.
    Section 3. ​
    Bill Joy was certainly intelligent, having achieved a perfect score on the math portion of his Scholastic Aptitude Test. His opportunities, however, were extraordinary. Bill Joy happened to choose a college that had a time-sharing computer available, something that was exceedingly rare during the early 1970s. While most students had to pay for computer time, Bill Joy exploited a bug in the college’s system to gain unlimited access to the Computer Center, which was walking distance from where he lived. He was able to spend almost all of his free time programming, and when he went to Berkeley, he had a terminal at home, where he could program until he fell asleep at his keyboard. Bill Joy was able to accumulate 10,000 hours while in college.
    Section 4. ​
    The Beatles are another example of the 10,000-hours principle. In 1960, when they were still a high school rock band, they were given the chance to travel to Hamburg, Germany and play in various strip clubs. The Beatles were contracted to play for hours every night, seven nights a week, and thereby were forced to develop as musicians. After a year and a half, they had performed 270 nights. By 1964, when they first found commercial success, they had performed an estimated twelve hundred times—more than most band bands do in an entire career.
    Section 5. ​
    Born in Seattle to affluent parents, Bill Gates was sent to a private school for seventh grade. The school’s Mother’s Club funded the purchase of a time-sharing terminal connected to a mainframe computer in Seattle. As an eighth-grader, Gates was able to start programming. When the Mother’s Club could no longer pay for computing time, Gates was able to continue programming at Information Sciences, Inc. (ISI) through a connection he made with one of the school parents. After that, one of ISI’s founders recommended Gates for a programming project at the Bonneville Power station. Gates was able to leave high school during his senior year to program for the Bonneville Power station as an independent study project. By the time Bill Gates dropped out of Harvard, he was well past 10,000 hours of experience.
    Section 6. ​
    In a list of the seventy-five richest people in human history, going as far back as ancient Egypt and converting all wealth estimates into modern U.S. dollars, fourteen names are Americans born within nine years of one another (including John D. Rockefeller and J.P. Morgan). What accounts for the fact that a list that covers thousands of years should have 20 percent of its members so close together? The fourteen Americans happened to live during the 1860s and 1870s, when the American economy was undergoing a huge transformation: the expansion of the railroads and the creation of Wall Street. Gladwell argues that the fourteen names are so close in birth date because they were all the perfect age to seize the opportunities available.
    Bill Joy and Bill Gates were also born at just the right time. In 1975, Popular Electronics ran a cover story about the Altair 8800, a $397 microcomputer kit. Someone born before 1954 would likely already have a job with IBM, working on mainframes. Someone born after 1956 would likely still be in high school. Most of the biggest pioneers in the modern computer industry, including Bill Gates, Bill Joy, Steve Jobs, and Eric Schmidt, were born in 1954 or 1955, as were Joy’s three fellow founders at Sun.
    Analysis: Chapter Two: The 10,000-Hour Rule ​
    Gladwell introduces his controversial and widely disputed 10,000 hours rule. Simply put, the rule suggests that to master a skill, an individual needs to practice it for 10,000 hours. Gladwell argues that, like hockey, success in computing requires both skill and luck, rather than pure, simple, natural ability. Gladwell's example is situated at the University of Michigan, specifically the new computer center, in 1971. The story of Bill Joy undermines the idea of success as an inexplicable, miraculous accomplishment. Though Gladwell agrees Joy is bright and talented, Joy also lucks into several happy accidents that allow for increased access. These strokes of luck give Joy the opportunity to become a computer expert. Gladwell's use of a psychological study conducted at the Academy of Music in Berlin, Germany, in the early 1990s forms the crux of his argument in this chapter. However, critics have suggested that this study was flawed and that there is no incontrovertible evidence to suggest that doing anything for 10,000 hours guarantees mastery. Rather, practice is, as Gladwell points out elsewhere throughout the text, just one of the factors in success. However, he also correlates the familial and economic advantages necessary to reach 10,000 of practice time, reinforcing the larger thesis of the book.
    Gladwell argues that the Beatles and Bill Gates are examples of his notion of time spent directly relating to mastery. While detailing how the Beatles reached the 10,000-hour threshold for mastery, Gladwell also highlights the lucky breaks they had in achieving success, like the connection that landed the band in Hamburg’s music scene. Additionally, while Gates has become something of an American folk hero, Gladwell argues that Gates did not reach his great achievements completely on his own. His wealthy family, his lucky interaction with a computer in 1968, and his college experience allowed him far greater access to opportunity. Without this combination of factors, along with his intelligence and drive, Gates would likely not have become a household name. Again, Gladwell reinforces the idea that success comes not only from being good, but from having no small amount of luck, whether that be time, money, or, in an echo of the hockey player example, birthdate.
    Gladwell claims that one important factor of success is the year in which one is born. In terms of wealth, there are 14 Americans, born in a nine-year period of the 1800s, who are among the richest people in history. This suggests that there was some connection to when they were born and the opportunities available to them in terms of innovation and invention, as well as the position they were in to take advantage of the opportunities. Similarly, men born in the same era of Bill Gates were coming of age at the bleeding edge of technology. Proximity to social and technological change, as well as the wealth and connections to take advantage of it, put these men in an advantageous position. Being in the right place at the right time, Gladwell argues, has at least as much to do with success as one's inborn abilities.
    Reference
    • “Outliers Chapter 2 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section2/. Accessed 27 Feb. 2023.
    ',20),r=[n];function s(h,l,c,d,u,m){return a(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
diff --git a/assets/save_reading_outliers_2.md.8febfe5f.lean.js b/assets/save_reading_outliers_2.md.ad1783d7.lean.js
similarity index 90%
rename from assets/save_reading_outliers_2.md.8febfe5f.lean.js
rename to assets/save_reading_outliers_2.md.ad1783d7.lean.js
index fe4d9c29..2141f71b 100644
--- a/assets/save_reading_outliers_2.md.8febfe5f.lean.js
+++ b/assets/save_reading_outliers_2.md.ad1783d7.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes","frontmatter":{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/2.md","filePath":"save/reading/outliers/2.md","lastUpdated":1684858945000}'),i={name:"save/reading/outliers/2.md"},n=o("",20),r=[n];function s(h,l,c,d,u,m){return a(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes","frontmatter":{"title":"Chapter Two: The 10,000-Hour Rule","description":"Summary of Chapter Two: The 10,000-Hour Rule of Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/2.md","filePath":"save/reading/outliers/2.md","lastUpdated":1684859551000}'),i={name:"save/reading/outliers/2.md"},n=o("",20),r=[n];function s(h,l,c,d,u,m){return a(),t("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
diff --git a/assets/save_reading_outliers_3.md.6e397ae9.js b/assets/save_reading_outliers_3.md.e6fcee78.js
similarity index 99%
rename from assets/save_reading_outliers_3.md.6e397ae9.js
rename to assets/save_reading_outliers_3.md.e6fcee78.js
index 774d578c..2508a9c7 100644
--- a/assets/save_reading_outliers_3.md.6e397ae9.js
+++ b/assets/save_reading_outliers_3.md.e6fcee78.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as s}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/3.md","filePath":"save/reading/outliers/3.md","lastUpdated":1684858945000}'),i={name:"save/reading/outliers/3.md"},o=s('
    Chapter 3: The Trouble with Geniuses, Part 1 ​
    Summary: Chapter Three: The Trouble with Geniuses, Part 1 ​
    Section 1. ​
    Chris Langan appeared on the television trivia show 1 vs. 100 after being the subject of interviews and at least one documentary, due to his extremely high IQ. He had little difficulty in high school, got a perfect score on his SAT, and read through Principia Mathematica as a teenager. On 1 vs. 100, Langan cashed out with winnings of $250,000.
    Section 2. ​
    Shortly after World War I, Lewis Terman, creator of the widely used Stanford-Binet IQ test, became fascinated with child prodigies. In 1921, he identified exceptionally gifted 1,470 students who came to be known as the “Termites.” Terman kept track of them over the years, carefully cataloging their educational achievement, marriages, psychological health, and employment. Terman was sure that the Termites would be the future elite of the United States. The same confidence underlies the use of IQ-like tests, such as the SAT, to assess applicants at Ivy League schools and corporations such as Microsoft and Google. Gladwell states that Terman was wrong about the Termites, and if he had met Chris Langan as a teenager, he would have been wrong about him, too.
    Section 3. ​
    Gladwell gives two examples—one easy and one much harder—of questions that use visual puzzles to test IQ, without requiring any language abilities or specific factual knowledge. The average human will score 100. Those scoring below 70 are considered mentally disabled, and those who are successful in graduate programs likely have an IQ of 115 or higher. According to Gladwell, however, above 120 IQ makes little or no difference to success. A similar point applies to basketball: height is certainly an advantage, but there is, at best, modest value in being six foot seven instead of six foot six. A basketball player just needs to be tall enough.
    A list of the last twenty-five Americans to win a Nobel Prize in Medicine includes graduates of prestigious schools, such as Columbia, Harvard, and MIT, but also graduates of DePauw, Holy Cross, and Gettysburg College. Similarly, there have been American Nobel laureates in Chemistry from Harvard and MIT, but also from Notre Dame and the University of Illinois. Harvard might have more intelligent students, but many other colleges have students smart enough to win a Nobel.
    At the University of Michigan law school, around 10 percent of students are racial minorities. Without affirmative action, there would be less than 3 percent. Looking at career performance after graduation, Michigan found no difference between white and minority graduates. Minority students come into law school with weaker academic credentials than their white counterparts, but they are above the threshold required to be just as successful.
    Section 4. ​
    Not all questions that test for intelligence have a single correct answer. A different sort of question might ask for a list of different uses for a brick, or a blanket. A question like this measures imagination and creativity. At a top British high school, one student lists more than ten uses, many of which are comical. Another student, a prodigy with one of the school’s highest IQs, lists just a few uses, all of which are practical. Compared to an IQ test, asking about different uses for a brick might be a better way to identify future Nobel Prize winners.
    Section 5. ​
    Terman’s error was using intelligence alone to choose his Termites. While some of the Termites had successful careers, an unexpectedly high went on to careers that even Terman didn’t consider to be successful. The two students tested by Terman’s team who actually did go on to win Nobels were not selected as Termites: their IQs were too low. Terman eventually concluded that “intellect and achievement are far from perfectly correlated.”
    Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.
    Analysis: Chapter Three: The Trouble with Geniuses, Part 1 ​
    Intelligence, Gladwell argues, does not equal success. He dedicates this chapter to the examination of geniuses. Comparing the young subjects of Lewis Terman to Chris Langan, Gladwell argues against Terman’s idea that genius is a key to societal advancement. Gladwell lists Langan’s early accomplishments as a shining example of Terman's idea, then swiftly undercuts expectations. Gladwell argues again that individuals do not succeed or fail solely on their own, but rather due to a constellation of factors, many of which are uncontrollable. Langan's early achievements would suggest great things and a continued path toward accomplishment, but Gladwell includes the anecdote of his game show appearance to undermine the notion that genius equals success. Using two questions from an IQ test, Gladwell introduces the idea of an IQ threshold to examine the the larger notion of intelligence and its limitations. Arguing that the measure of someone’s intelligence is only helpful up to a point, Gladwell uses examples of schools that Nobel laureates attended for undergraduate studies. Some, he points out, are elite institutions; others are not. Additionally, he points to success among graduates of the law school at the University of Michigan. There was no quantifiable difference in success post-graduation between white and POC students, some of whom were admitted under affirmative action with slightly lower undergraduate grades and standardized test scores. Reasonably, then, the level of intelligence beyond a certain point is moot. All of the students were intelligent enough to graduate, and beyond that, he argues, their intelligence predicted nothing.
    Elements other than intelligence must contribute to success. Gladwell expands on his notion of an intelligence threshold and examines these other factors through an example of a question on a divergence test. Using examples of two students at an elite British school, Gladwell shows that the divergence test has no right answers, but instead looks for answers that demonstrate uniqueness and creativity. Intelligence matters, but so does imagination, creativity, and outside-the-box thinking. Gladwell circles back to Terman, reinforcing that the error when choosing his Termites was a focus on intelligence alone. He once again turns his attention to Chris Langan, foreshadowing that Langan’s story did not follow a direct line to achievement. Intelligence alone does not offer a certain path to success.
    Reference
    • “Outliers Chapter 3 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section3/. Accessed 27 Feb. 2023.
    ',19),n=[o];function r(l,h,c,u,d,m){return a(),t("div",null,n)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
+import{_ as e,c as t,o as a,R as s}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/3.md","filePath":"save/reading/outliers/3.md","lastUpdated":1684859551000}'),i={name:"save/reading/outliers/3.md"},o=s('
    Chapter 3: The Trouble with Geniuses, Part 1 ​
    Summary: Chapter Three: The Trouble with Geniuses, Part 1 ​
    Section 1. ​
    Chris Langan appeared on the television trivia show 1 vs. 100 after being the subject of interviews and at least one documentary, due to his extremely high IQ. He had little difficulty in high school, got a perfect score on his SAT, and read through Principia Mathematica as a teenager. On 1 vs. 100, Langan cashed out with winnings of $250,000.
    Section 2. ​
    Shortly after World War I, Lewis Terman, creator of the widely used Stanford-Binet IQ test, became fascinated with child prodigies. In 1921, he identified exceptionally gifted 1,470 students who came to be known as the “Termites.” Terman kept track of them over the years, carefully cataloging their educational achievement, marriages, psychological health, and employment. Terman was sure that the Termites would be the future elite of the United States. The same confidence underlies the use of IQ-like tests, such as the SAT, to assess applicants at Ivy League schools and corporations such as Microsoft and Google. Gladwell states that Terman was wrong about the Termites, and if he had met Chris Langan as a teenager, he would have been wrong about him, too.
    Section 3. ​
    Gladwell gives two examples—one easy and one much harder—of questions that use visual puzzles to test IQ, without requiring any language abilities or specific factual knowledge. The average human will score 100. Those scoring below 70 are considered mentally disabled, and those who are successful in graduate programs likely have an IQ of 115 or higher. According to Gladwell, however, above 120 IQ makes little or no difference to success. A similar point applies to basketball: height is certainly an advantage, but there is, at best, modest value in being six foot seven instead of six foot six. A basketball player just needs to be tall enough.
    A list of the last twenty-five Americans to win a Nobel Prize in Medicine includes graduates of prestigious schools, such as Columbia, Harvard, and MIT, but also graduates of DePauw, Holy Cross, and Gettysburg College. Similarly, there have been American Nobel laureates in Chemistry from Harvard and MIT, but also from Notre Dame and the University of Illinois. Harvard might have more intelligent students, but many other colleges have students smart enough to win a Nobel.
    At the University of Michigan law school, around 10 percent of students are racial minorities. Without affirmative action, there would be less than 3 percent. Looking at career performance after graduation, Michigan found no difference between white and minority graduates. Minority students come into law school with weaker academic credentials than their white counterparts, but they are above the threshold required to be just as successful.
    Section 4. ​
    Not all questions that test for intelligence have a single correct answer. A different sort of question might ask for a list of different uses for a brick, or a blanket. A question like this measures imagination and creativity. At a top British high school, one student lists more than ten uses, many of which are comical. Another student, a prodigy with one of the school’s highest IQs, lists just a few uses, all of which are practical. Compared to an IQ test, asking about different uses for a brick might be a better way to identify future Nobel Prize winners.
    Section 5. ​
    Terman’s error was using intelligence alone to choose his Termites. While some of the Termites had successful careers, an unexpectedly high went on to careers that even Terman didn’t consider to be successful. The two students tested by Terman’s team who actually did go on to win Nobels were not selected as Termites: their IQs were too low. Terman eventually concluded that “intellect and achievement are far from perfectly correlated.”
    Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.
    Analysis: Chapter Three: The Trouble with Geniuses, Part 1 ​
    Intelligence, Gladwell argues, does not equal success. He dedicates this chapter to the examination of geniuses. Comparing the young subjects of Lewis Terman to Chris Langan, Gladwell argues against Terman’s idea that genius is a key to societal advancement. Gladwell lists Langan’s early accomplishments as a shining example of Terman's idea, then swiftly undercuts expectations. Gladwell argues again that individuals do not succeed or fail solely on their own, but rather due to a constellation of factors, many of which are uncontrollable. Langan's early achievements would suggest great things and a continued path toward accomplishment, but Gladwell includes the anecdote of his game show appearance to undermine the notion that genius equals success. Using two questions from an IQ test, Gladwell introduces the idea of an IQ threshold to examine the the larger notion of intelligence and its limitations. Arguing that the measure of someone’s intelligence is only helpful up to a point, Gladwell uses examples of schools that Nobel laureates attended for undergraduate studies. Some, he points out, are elite institutions; others are not. Additionally, he points to success among graduates of the law school at the University of Michigan. There was no quantifiable difference in success post-graduation between white and POC students, some of whom were admitted under affirmative action with slightly lower undergraduate grades and standardized test scores. Reasonably, then, the level of intelligence beyond a certain point is moot. All of the students were intelligent enough to graduate, and beyond that, he argues, their intelligence predicted nothing.
    Elements other than intelligence must contribute to success. Gladwell expands on his notion of an intelligence threshold and examines these other factors through an example of a question on a divergence test. Using examples of two students at an elite British school, Gladwell shows that the divergence test has no right answers, but instead looks for answers that demonstrate uniqueness and creativity. Intelligence matters, but so does imagination, creativity, and outside-the-box thinking. Gladwell circles back to Terman, reinforcing that the error when choosing his Termites was a focus on intelligence alone. He once again turns his attention to Chris Langan, foreshadowing that Langan’s story did not follow a direct line to achievement. Intelligence alone does not offer a certain path to success.
    Reference
    • “Outliers Chapter 3 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section3/. Accessed 27 Feb. 2023.
    ',19),n=[o];function r(l,h,c,u,d,m){return a(),t("div",null,n)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
diff --git a/assets/save_reading_outliers_3.md.6e397ae9.lean.js b/assets/save_reading_outliers_3.md.e6fcee78.lean.js
similarity index 90%
rename from assets/save_reading_outliers_3.md.6e397ae9.lean.js
rename to assets/save_reading_outliers_3.md.e6fcee78.lean.js
index 16839eeb..580b4a1b 100644
--- a/assets/save_reading_outliers_3.md.6e397ae9.lean.js
+++ b/assets/save_reading_outliers_3.md.e6fcee78.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as s}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/3.md","filePath":"save/reading/outliers/3.md","lastUpdated":1684858945000}'),i={name:"save/reading/outliers/3.md"},o=s("",19),n=[o];function r(l,h,c,u,d,m){return a(),t("div",null,n)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
+import{_ as e,c as t,o as a,R as s}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 3: The Trouble with Geniuses, Part 1","description":"Chapter 3 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/3.md","filePath":"save/reading/outliers/3.md","lastUpdated":1684859551000}'),i={name:"save/reading/outliers/3.md"},o=s("",19),n=[o];function r(l,h,c,u,d,m){return a(),t("div",null,n)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
diff --git a/assets/save_reading_outliers_4.md.a4a3d333.js b/assets/save_reading_outliers_4.md.62591abe.js
similarity index 99%
rename from assets/save_reading_outliers_4.md.a4a3d333.js
rename to assets/save_reading_outliers_4.md.62591abe.js
index 348cbaa5..47c6cc2f 100644
--- a/assets/save_reading_outliers_4.md.a4a3d333.js
+++ b/assets/save_reading_outliers_4.md.62591abe.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/4.md","filePath":"save/reading/outliers/4.md","lastUpdated":1684858945000}'),n={name:"save/reading/outliers/4.md"},i=o('
    Chapter 4: The Trouble with Geniuses, Part 2 ​
    Summary: Chapter Four: The Trouble with Geniuses, Part 2 ​
    Section 1. ​
    Chris Langan’s mother had four children, all from different fathers. Her first husband died in Mexico, the second was murdered, the third committed suicide, and the fourth was an abusive alcoholic. Langan says he has never met someone who grew up as impoverished as he and his brothers did. Attending Reed College, in Oregon, on a scholarship, he felt as though he did not belong with the rest of the students. When his mother neglected to fill out the financial aid forms, he lost his scholarship and dropped out. According to Langan, the school just did not care about their students. Later, Langan attended Montana State University in Bozeman but he dropped out when a dean refused Langan’s request to adjust his class schedule because of car trouble. He still has intellectual interests and has been working on a treatise called the “Cognitive Theoretic Model of the Universe.” When asked if, hypothetically, he would take a position at Harvard, he acknowledges the benefit of an environment with so much intellectual energy, but he worries about a lack of intellectual freedom.
    Section 2. ​
    Gladwell points out the oddity of Langan’s experiences. Most colleges, especially small ones, try to accommodate student needs. Smart people take positions at places like Harvard instead of in the private sector precisely for the added intellectual freedom. Gladwell contrasts Langan’s story with Robert Oppenheimer’s. While at Cambridge, Oppenheimer became frustrated with his tutor and tried to poison him. After negotiations with the school administration, Oppenheimer was put on probation and assigned to a psychiatrist. Whereas Langan dropped out after his mother failed to fill out paperwork, Oppenheimer managed to stay in college after trying to kill someone. Later, Oppenheimer convinced General Leslie Groves to make Oppenheimer the scientific director of the Manhattan Project. Oppenheimer had the ability to persuade people, where Langan did not.
    Section 3. ​
    Psychologist Robert Sternberg calls the ability to persuade someone “practical intelligence.” It involves knowing what to say, how to say it, and when. It requires an ability to read situations. Individuals can have either practical or analytical intelligence, or in the case of Oppenheimer, both. To understand where one obtains practical intelligence, Gladwell refers to a study conducted by Annette Lareau.
    Lareau observed parenting strategies in both Black and white families from different socioeconomic strata. She found two distinct strategies: “concerted cultivation” and “accomplishment of natural growth.” Concerted cultivation, most common in middle-class and wealthier families, is defined by parents’ active promotion of their children’s talents and interests. Accomplishment of natural growth, most common in working-class or impoverished families, is defined by parents’ lack of interest or interaction in their children’s free time. Concerted cultivation, writes Gladwell, encourages children to seek opportunities for growth and to advocate for themselves. Accomplishment of natural growth leaves children unprepared to shape their environment for their own benefit. Concerted cultivation fosters an attitude better suited for success in the modern world.
    Section 4. ​
    Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.
    • Affluent: Having an abundant supply of money or possessions of value.
    By contrast, Chris Langan’s brother Mark states that as children, they learned to resent authority. Mark, too, was unable to obtain financial aid, because he had no knowledge of how the system worked. If Chris Langan had been raised in a family that valued education, he probably would have had more success.
    Section 5. ​
    Once the Termites reached working adulthood, Terman grouped 730 of the males into three categories, based on their levels of success. The top 20 percent were upper-class professionals, while the bottom 20 percent had low-paying jobs or were unemployed. After analyzing the differences between the groups, Terman found that the bottom 20 percent came almost entirely from the lowest social and economic classes. What they lacked was “a community around them that prepared them properly for the world.”
    Section 6. ​
    Chris Langan lives on a horse farm in Missouri, where he still writes. He is happily married and continues to read physics and philosophy. While he has spent decades on his writing, almost none of it has ever been published. Langan admits he has not tried to contact publishers or found an agent—and does not plan to. Chris Langan did not have the support necessary to meet success. Gladwell points that no one ever makes it alone, regardless of if they are successful musicians, athletes, tech billionaires, or geniuses.
    Analysis: Chapter Four: The Trouble with Geniuses, Part 2 ​
    Family background plays a crucial role in an individual’s success. While a genius like Chris Langan should thrive on a college campus, he drops out. Langan doesn’t struggle with academics, but with paperwork, procedures, and environment. Gladwell’s characterization of Langan almost makes him seem like two separate characters: a confident, genius game show contestant versus a child from a poor and troubled family. Langan may be a true outlier in the sense that he doesn’t seem to fit in academia or the working-class world.  Langan’s story is ironic. His brilliant mind should propel him to success. However, his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success. The author delves deeper into the irony of Langan’s situation by comparing him to Robert Oppenheimer, who worked on the nuclear bomb during World War II. Like Langan, Oppenheimer possesses great intelligence. Unlike Langan, Oppenheimer has an understanding that allows him to navigate the world effectively. Oppenheimer knows how to combat challenges, where Langan gets bested by a financial aid problem and class schedule. Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority. Without a family background that encourages success, intelligence is not enough.
    Contrast & Comparison
    Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.
    The power of persuasion is an important factor that drives success. Gladwell notes that the power of persuasion is practical intelligence. He contrasts practical intelligence with analytical ability, noting that just because a person is smart does not mean they have practical intelligence. These two types of knowledge are not necessarily coeval. He reinforces this point with a return to the comparison of Langan and Oppenheimer. Both men are exceedingly intelligent, but Langan lacks the social savvy of practical intelligence. Gladwell suggests that individuals gain practical intelligence skills from their families, discussing a study by sociologist Annette Lareau to prove his point. Children from wealthier families learn how to interact with society in a way that drives achievement. Successful people have the power of persuasion and can convince others to see things from their point of view.
    Success, then, depends on more than individual effort. Gladwell builds this argument by detailing the differences between Oppenheimer’s upbringing and Langan’s. Oppenheimer lived in an upper-class neighborhood with myriad advantages. He learned how to negotiate and stand up for himself. Langan’s family moved from place to place, struggling for jobs, money, food, and clothes, and Langan learned that he couldn’t rely on others. Gladwell's consideration of the Termites' success or struggles as adults supports his assertion that success is more than an individual effort; Gladwell states, “In the end, only one thing mattered: family background.” Terman’s findings and Gladwell’s conclusion do not mean that talented individuals from lower social and economic classes cannot have successful futures. To succeed, everyone needs help along the way, if not from family, then from a supportive community. Individuals do not achieve success solely on their own.
    Reference
    • “Outliers Chapter 4 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section4/.
    ',23),s=[i];function r(l,h,c,d,u,p){return a(),t("div",null,s)}const g=e(n,[["render",r]]);export{f as __pageData,g as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/4.md","filePath":"save/reading/outliers/4.md","lastUpdated":1684859551000}'),n={name:"save/reading/outliers/4.md"},i=o('
    Chapter 4: The Trouble with Geniuses, Part 2 ​
    Summary: Chapter Four: The Trouble with Geniuses, Part 2 ​
    Section 1. ​
    Chris Langan’s mother had four children, all from different fathers. Her first husband died in Mexico, the second was murdered, the third committed suicide, and the fourth was an abusive alcoholic. Langan says he has never met someone who grew up as impoverished as he and his brothers did. Attending Reed College, in Oregon, on a scholarship, he felt as though he did not belong with the rest of the students. When his mother neglected to fill out the financial aid forms, he lost his scholarship and dropped out. According to Langan, the school just did not care about their students. Later, Langan attended Montana State University in Bozeman but he dropped out when a dean refused Langan’s request to adjust his class schedule because of car trouble. He still has intellectual interests and has been working on a treatise called the “Cognitive Theoretic Model of the Universe.” When asked if, hypothetically, he would take a position at Harvard, he acknowledges the benefit of an environment with so much intellectual energy, but he worries about a lack of intellectual freedom.
    Section 2. ​
    Gladwell points out the oddity of Langan’s experiences. Most colleges, especially small ones, try to accommodate student needs. Smart people take positions at places like Harvard instead of in the private sector precisely for the added intellectual freedom. Gladwell contrasts Langan’s story with Robert Oppenheimer’s. While at Cambridge, Oppenheimer became frustrated with his tutor and tried to poison him. After negotiations with the school administration, Oppenheimer was put on probation and assigned to a psychiatrist. Whereas Langan dropped out after his mother failed to fill out paperwork, Oppenheimer managed to stay in college after trying to kill someone. Later, Oppenheimer convinced General Leslie Groves to make Oppenheimer the scientific director of the Manhattan Project. Oppenheimer had the ability to persuade people, where Langan did not.
    Section 3. ​
    Psychologist Robert Sternberg calls the ability to persuade someone “practical intelligence.” It involves knowing what to say, how to say it, and when. It requires an ability to read situations. Individuals can have either practical or analytical intelligence, or in the case of Oppenheimer, both. To understand where one obtains practical intelligence, Gladwell refers to a study conducted by Annette Lareau.
    Lareau observed parenting strategies in both Black and white families from different socioeconomic strata. She found two distinct strategies: “concerted cultivation” and “accomplishment of natural growth.” Concerted cultivation, most common in middle-class and wealthier families, is defined by parents’ active promotion of their children’s talents and interests. Accomplishment of natural growth, most common in working-class or impoverished families, is defined by parents’ lack of interest or interaction in their children’s free time. Concerted cultivation, writes Gladwell, encourages children to seek opportunities for growth and to advocate for themselves. Accomplishment of natural growth leaves children unprepared to shape their environment for their own benefit. Concerted cultivation fosters an attitude better suited for success in the modern world.
    Section 4. ​
    Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.
    • Affluent: Having an abundant supply of money or possessions of value.
    By contrast, Chris Langan’s brother Mark states that as children, they learned to resent authority. Mark, too, was unable to obtain financial aid, because he had no knowledge of how the system worked. If Chris Langan had been raised in a family that valued education, he probably would have had more success.
    Section 5. ​
    Once the Termites reached working adulthood, Terman grouped 730 of the males into three categories, based on their levels of success. The top 20 percent were upper-class professionals, while the bottom 20 percent had low-paying jobs or were unemployed. After analyzing the differences between the groups, Terman found that the bottom 20 percent came almost entirely from the lowest social and economic classes. What they lacked was “a community around them that prepared them properly for the world.”
    Section 6. ​
    Chris Langan lives on a horse farm in Missouri, where he still writes. He is happily married and continues to read physics and philosophy. While he has spent decades on his writing, almost none of it has ever been published. Langan admits he has not tried to contact publishers or found an agent—and does not plan to. Chris Langan did not have the support necessary to meet success. Gladwell points that no one ever makes it alone, regardless of if they are successful musicians, athletes, tech billionaires, or geniuses.
    Analysis: Chapter Four: The Trouble with Geniuses, Part 2 ​
    Family background plays a crucial role in an individual’s success. While a genius like Chris Langan should thrive on a college campus, he drops out. Langan doesn’t struggle with academics, but with paperwork, procedures, and environment. Gladwell’s characterization of Langan almost makes him seem like two separate characters: a confident, genius game show contestant versus a child from a poor and troubled family. Langan may be a true outlier in the sense that he doesn’t seem to fit in academia or the working-class world.  Langan’s story is ironic. His brilliant mind should propel him to success. However, his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success. The author delves deeper into the irony of Langan’s situation by comparing him to Robert Oppenheimer, who worked on the nuclear bomb during World War II. Like Langan, Oppenheimer possesses great intelligence. Unlike Langan, Oppenheimer has an understanding that allows him to navigate the world effectively. Oppenheimer knows how to combat challenges, where Langan gets bested by a financial aid problem and class schedule. Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority. Without a family background that encourages success, intelligence is not enough.
    Contrast & Comparison
    Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.
    The power of persuasion is an important factor that drives success. Gladwell notes that the power of persuasion is practical intelligence. He contrasts practical intelligence with analytical ability, noting that just because a person is smart does not mean they have practical intelligence. These two types of knowledge are not necessarily coeval. He reinforces this point with a return to the comparison of Langan and Oppenheimer. Both men are exceedingly intelligent, but Langan lacks the social savvy of practical intelligence. Gladwell suggests that individuals gain practical intelligence skills from their families, discussing a study by sociologist Annette Lareau to prove his point. Children from wealthier families learn how to interact with society in a way that drives achievement. Successful people have the power of persuasion and can convince others to see things from their point of view.
    Success, then, depends on more than individual effort. Gladwell builds this argument by detailing the differences between Oppenheimer’s upbringing and Langan’s. Oppenheimer lived in an upper-class neighborhood with myriad advantages. He learned how to negotiate and stand up for himself. Langan’s family moved from place to place, struggling for jobs, money, food, and clothes, and Langan learned that he couldn’t rely on others. Gladwell's consideration of the Termites' success or struggles as adults supports his assertion that success is more than an individual effort; Gladwell states, “In the end, only one thing mattered: family background.” Terman’s findings and Gladwell’s conclusion do not mean that talented individuals from lower social and economic classes cannot have successful futures. To succeed, everyone needs help along the way, if not from family, then from a supportive community. Individuals do not achieve success solely on their own.
    Reference
    • “Outliers Chapter 4 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section4/.
    ',23),s=[i];function r(l,h,c,d,u,p){return a(),t("div",null,s)}const g=e(n,[["render",r]]);export{f as __pageData,g as default};
diff --git a/assets/save_reading_outliers_4.md.a4a3d333.lean.js b/assets/save_reading_outliers_4.md.62591abe.lean.js
similarity index 90%
rename from assets/save_reading_outliers_4.md.a4a3d333.lean.js
rename to assets/save_reading_outliers_4.md.62591abe.lean.js
index 5ca60022..2246bf71 100644
--- a/assets/save_reading_outliers_4.md.a4a3d333.lean.js
+++ b/assets/save_reading_outliers_4.md.62591abe.lean.js
@@ -1 +1 @@
-import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/4.md","filePath":"save/reading/outliers/4.md","lastUpdated":1684858945000}'),n={name:"save/reading/outliers/4.md"},i=o("",23),s=[i];function r(l,h,c,d,u,p){return a(),t("div",null,s)}const g=e(n,[["render",r]]);export{f as __pageData,g as default};
+import{_ as e,c as t,o as a,R as o}from"./chunks/framework.a1bac067.js";const f=JSON.parse('{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes","frontmatter":{"title":"Chapter 4: The Trouble with Geniuses, Part 2","description":"Chapter 4 summary of novel Outliers from Sparknotes"},"headers":[],"relativePath":"save/reading/outliers/4.md","filePath":"save/reading/outliers/4.md","lastUpdated":1684859551000}'),n={name:"save/reading/outliers/4.md"},i=o("",23),s=[i];function r(l,h,c,d,u,p){return a(),t("div",null,s)}const g=e(n,[["render",r]]);export{f as __pageData,g as default};
diff --git a/assets/style.0b53ab5c.css b/assets/style.250ef51a.css
similarity index 62%
rename from assets/style.0b53ab5c.css
rename to assets/style.250ef51a.css
index 66184284..b9cd006e 100644
--- a/assets/style.0b53ab5c.css
+++ b/assets/style.250ef51a.css
@@ -1 +1 @@
-@charset "UTF-8";@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-cyrillic.5f2c6c8c.woff2) format("woff2");unicode-range:U+0301,U+0400-045F,U+0490-0491,U+04B0-04B1,U+2116}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-cyrillic-ext.e75737ce.woff2) format("woff2");unicode-range:U+0460-052F,U+1C80-1C88,U+20B4,U+2DE0-2DFF,U+A640-A69F,U+FE2E-FE2F}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-greek.d5a6d92a.woff2) format("woff2");unicode-range:U+0370-03FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-greek-ext.ab0619bc.woff2) format("woff2");unicode-range:U+1F00-1FFF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-latin.2ed14f66.woff2) format("woff2");unicode-range:U+0000-00FF,U+0131,U+0152-0153,U+02BB-02BC,U+02C6,U+02DA,U+02DC,U+2000-206F,U+2074,U+20AC,U+2122,U+2191,U+2193,U+2212,U+2215,U+FEFF,U+FFFD}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-latin-ext.0030eebd.woff2) format("woff2");unicode-range:U+0100-024F,U+0259,U+1E00-1EFF,U+2020,U+20A0-20AB,U+20AD-20CF,U+2113,U+2C60-2C7F,U+A720-A7FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-vietnamese.14ce25a6.woff2) format("woff2");unicode-range:U+0102-0103,U+0110-0111,U+0128-0129,U+0168-0169,U+01A0-01A1,U+01AF-01B0,U+1EA0-1EF9,U+20AB}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-cyrillic.ea42a392.woff2) format("woff2");unicode-range:U+0301,U+0400-045F,U+0490-0491,U+04B0-04B1,U+2116}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-cyrillic-ext.33bd5a8e.woff2) format("woff2");unicode-range:U+0460-052F,U+1C80-1C88,U+20B4,U+2DE0-2DFF,U+A640-A69F,U+FE2E-FE2F}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-greek.8f4463c4.woff2) format("woff2");unicode-range:U+0370-03FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-greek-ext.4fbe9427.woff2) format("woff2");unicode-range:U+1F00-1FFF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-latin.bd3b6f56.woff2) format("woff2");unicode-range:U+0000-00FF,U+0131,U+0152-0153,U+02BB-02BC,U+02C6,U+02DA,U+02DC,U+2000-206F,U+2074,U+20AC,U+2122,U+2191,U+2193,U+2212,U+2215,U+FEFF,U+FFFD}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-latin-ext.bd8920cc.woff2) format("woff2");unicode-range:U+0100-024F,U+0259,U+1E00-1EFF,U+2020,U+20A0-20AB,U+20AD-20CF,U+2113,U+2C60-2C7F,U+A720-A7FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-vietnamese.6ce511fb.woff2) format("woff2");unicode-range:U+0102-0103,U+0110-0111,U+0128-0129,U+0168-0169,U+01A0-01A1,U+01AF-01B0,U+1EA0-1EF9,U+20AB}@font-face{font-family:Chinese Quotes;src:local("PingFang SC Regular"),local("PingFang SC"),local("SimHei"),local("Source Han Sans SC");unicode-range:U+2018,U+2019,U+201C,U+201D}:root{--vp-c-white: #ffffff;--vp-c-black: #000000;--vp-c-gray: #8e8e93;--vp-c-text-light-1: rgba(60, 60, 67);--vp-c-text-light-2: rgba(60, 60, 67, .75);--vp-c-text-light-3: rgba(60, 60, 67, .33);--vp-c-text-dark-1: rgba(255, 255, 245, .86);--vp-c-text-dark-2: rgba(235, 235, 245, .6);--vp-c-text-dark-3: rgba(235, 235, 245, .38);--vp-c-green: #10b981;--vp-c-green-light: #34d399;--vp-c-green-lighter: #6ee7b7;--vp-c-green-dark: #059669;--vp-c-green-darker: #047857;--vp-c-green-dimm-1: rgba(16, 185, 129, .05);--vp-c-green-dimm-2: rgba(16, 185, 129, .2);--vp-c-green-dimm-3: rgba(16, 185, 129, .5);--vp-c-yellow: #d97706;--vp-c-yellow-light: #f59e0b;--vp-c-yellow-lighter: #fbbf24;--vp-c-yellow-dark: #b45309;--vp-c-yellow-darker: #92400e;--vp-c-yellow-dimm-1: rgba(234, 179, 8, .05);--vp-c-yellow-dimm-2: rgba(234, 179, 8, .2);--vp-c-yellow-dimm-3: rgba(234, 179, 8, .5);--vp-c-red: #f43f5e;--vp-c-red-light: #fb7185;--vp-c-red-lighter: #fda4af;--vp-c-red-dark: #e11d48;--vp-c-red-darker: #be123c;--vp-c-red-dimm-1: rgba(244, 63, 94, .05);--vp-c-red-dimm-2: rgba(244, 63, 94, .2);--vp-c-red-dimm-3: rgba(244, 63, 94, .5);--vp-c-sponsor: #db2777}:root{--vp-c-bg: #ffffff;--vp-c-bg-elv: #ffffff;--vp-c-bg-elv-up: #ffffff;--vp-c-bg-elv-down: #f6f6f7;--vp-c-bg-elv-mute: #f6f6f7;--vp-c-bg-soft: #f6f6f7;--vp-c-bg-soft-up: #f9f9fa;--vp-c-bg-soft-down: #e3e3e5;--vp-c-bg-soft-mute: #e3e3e5;--vp-c-bg-alt: #f6f6f7;--vp-c-border: rgba(60, 60, 67, .29);--vp-c-divider: rgba(60, 60, 67, .12);--vp-c-gutter: rgba(60, 60, 67, .12);--vp-c-neutral: var(--vp-c-black);--vp-c-neutral-inverse: var(--vp-c-white);--vp-c-text-1: var(--vp-c-text-light-1);--vp-c-text-2: var(--vp-c-text-light-2);--vp-c-text-3: var(--vp-c-text-light-3);--vp-c-text-inverse-1: var(--vp-c-text-dark-1);--vp-c-text-inverse-2: var(--vp-c-text-dark-2);--vp-c-text-inverse-3: var(--vp-c-text-dark-3);--vp-c-text-code: #476582;--vp-c-brand: var(--vp-c-green);--vp-c-brand-light: var(--vp-c-green-light);--vp-c-brand-lighter: var(--vp-c-green-lighter);--vp-c-brand-dark: var(--vp-c-green-dark);--vp-c-brand-darker: var(--vp-c-green-darker);--vp-c-mute: #f6f6f7;--vp-c-mute-light: #f9f9fc;--vp-c-mute-lighter: #ffffff;--vp-c-mute-dark: #e3e3e5;--vp-c-mute-darker: #d7d7d9}.dark{--vp-c-bg: #1e1e20;--vp-c-bg-elv: #252529;--vp-c-bg-elv-up: #313136;--vp-c-bg-elv-down: #1e1e20;--vp-c-bg-elv-mute: #313136;--vp-c-bg-soft: #252529;--vp-c-bg-soft-up: #313136;--vp-c-bg-soft-down: #1e1e20;--vp-c-bg-soft-mute: #313136;--vp-c-bg-alt: #161618;--vp-c-border: rgba(82, 82, 89, .68);--vp-c-divider: rgba(82, 82, 89, .32);--vp-c-gutter: #000000;--vp-c-neutral: var(--vp-c-white);--vp-c-neutral-inverse: var(--vp-c-black);--vp-c-text-1: var(--vp-c-text-dark-1);--vp-c-text-2: var(--vp-c-text-dark-2);--vp-c-text-3: var(--vp-c-text-dark-3);--vp-c-text-inverse-1: var(--vp-c-text-light-1);--vp-c-text-inverse-2: var(--vp-c-text-light-2);--vp-c-text-inverse-3: var(--vp-c-text-light-3);--vp-c-text-code: #c9def1;--vp-c-mute: #313136;--vp-c-mute-light: #3a3a3c;--vp-c-mute-lighter: #505053;--vp-c-mute-dark: #2c2c30;--vp-c-mute-darker: #252529}:root{--vp-font-family-base: "Chinese Quotes", "Inter var", "Inter", ui-sans-serif, system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Helvetica, Arial, "Noto Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";--vp-font-family-mono: ui-monospace, SFMono-Regular, "SF Mono", Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace}:root{--vp-shadow-1: 0 1px 2px rgba(0, 0, 0, .04), 0 1px 2px rgba(0, 0, 0, .06);--vp-shadow-2: 0 3px 12px rgba(0, 0, 0, .07), 0 1px 4px rgba(0, 0, 0, .07);--vp-shadow-3: 0 12px 32px rgba(0, 0, 0, .1), 0 2px 6px rgba(0, 0, 0, .08);--vp-shadow-4: 0 14px 44px rgba(0, 0, 0, .12), 0 3px 9px rgba(0, 0, 0, .12);--vp-shadow-5: 0 18px 56px rgba(0, 0, 0, .16), 0 4px 12px rgba(0, 0, 0, .16)}:root{--vp-z-index-local-nav: 10;--vp-z-index-nav: 20;--vp-z-index-layout-top: 30;--vp-z-index-backdrop: 40;--vp-z-index-sidebar: 50;--vp-z-index-footer: 60}:root{--vp-icon-copy: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' height='20' width='20' stroke='rgba(128,128,128,1)' stroke-width='2' viewBox='0 0 24 24'%3E%3Cpath stroke-linecap='round' stroke-linejoin='round' d='M9 5H7a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h10a2 2 0 0 0 2-2V7a2 2 0 0 0-2-2h-2M9 5a2 2 0 0 0 2 2h2a2 2 0 0 0 2-2M9 5a2 2 0 0 1 2-2h2a2 2 0 0 1 2 2'/%3E%3C/svg%3E");--vp-icon-copied: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' height='20' width='20' stroke='rgba(128,128,128,1)' stroke-width='2' viewBox='0 0 24 24'%3E%3Cpath stroke-linecap='round' stroke-linejoin='round' d='M9 5H7a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h10a2 2 0 0 0 2-2V7a2 2 0 0 0-2-2h-2M9 5a2 2 0 0 0 2 2h2a2 2 0 0 0 2-2M9 5a2 2 0 0 1 2-2h2a2 2 0 0 1 2 2m-6 9 2 2 4-4'/%3E%3C/svg%3E")}:root{--vp-layout-max-width: 1440px}:root{--vp-header-anchor-symbol: "#"}:root{--vp-code-line-height: 1.7;--vp-code-font-size: .875em;--vp-c-code-dimm: var(--vp-c-text-dark-3);--vp-code-block-color: var(--vp-c-text-dark-1);--vp-code-block-bg: #292b30;--vp-code-block-bg-light: #1e1e20;--vp-code-block-divider-color: #000000;--vp-code-line-highlight-color: rgba(0, 0, 0, .5);--vp-code-line-number-color: var(--vp-c-code-dimm);--vp-code-line-diff-add-color: var(--vp-c-green-dimm-2);--vp-code-line-diff-add-symbol-color: var(--vp-c-green);--vp-code-line-diff-remove-color: var(--vp-c-red-dimm-2);--vp-code-line-diff-remove-symbol-color: var(--vp-c-red);--vp-code-line-warning-color: var(--vp-c-yellow-dimm-2);--vp-code-line-error-color: var(--vp-c-red-dimm-2);--vp-code-copy-code-border-color: transparent;--vp-code-copy-code-bg: var(--vp-code-block-bg-light);--vp-code-copy-code-hover-border-color: var(--vp-c-divider);--vp-code-copy-code-hover-bg: var(--vp-code-block-bg-light);--vp-code-copy-code-active-text: var(--vp-c-text-dark-2);--vp-code-tab-divider: var(--vp-code-block-divider-color);--vp-code-tab-text-color: var(--vp-c-text-dark-2);--vp-code-tab-bg: var(--vp-code-block-bg);--vp-code-tab-hover-text-color: var(--vp-c-text-dark-1);--vp-code-tab-active-text-color: var(--vp-c-text-dark-1);--vp-code-tab-active-bar-color: var(--vp-c-brand)}.dark{--vp-code-block-bg: #161618}:root:not(.dark) .vp-adaptive-theme{--vp-c-code-dimm: var(--vp-c-text-2);--vp-code-block-color: var(--vp-c-text-1);--vp-code-block-bg: #f8f8f8;--vp-code-block-divider-color: var(--vp-c-divider);--vp-code-line-highlight-color: #ececec;--vp-code-line-number-color: var(--vp-c-code-dimm);--vp-code-copy-code-bg: #e2e2e2;--vp-code-copy-code-hover-bg: #dcdcdc;--vp-code-copy-code-active-text: var(--vp-c-text-2);--vp-code-tab-divider: var(--vp-c-divider);--vp-code-tab-text-color: var(--vp-c-text-2);--vp-code-tab-bg: var(--vp-code-block-bg);--vp-code-tab-hover-text-color: var(--vp-c-text-1);--vp-code-tab-active-text-color: var(--vp-c-text-1)}:root{--vp-button-brand-border: var(--vp-c-brand-lighter);--vp-button-brand-text: var(--vp-c-white);--vp-button-brand-bg: var(--vp-c-brand);--vp-button-brand-hover-border: var(--vp-c-brand-lighter);--vp-button-brand-hover-text: var(--vp-c-white);--vp-button-brand-hover-bg: var(--vp-c-brand-dark);--vp-button-brand-active-border: var(--vp-c-brand-lighter);--vp-button-brand-active-text: var(--vp-c-white);--vp-button-brand-active-bg: var(--vp-c-brand-darker);--vp-button-alt-border: var(--vp-c-border);--vp-button-alt-text: var(--vp-c-neutral);--vp-button-alt-bg: var(--vp-c-mute);--vp-button-alt-hover-border: var(--vp-c-border);--vp-button-alt-hover-text: var(--vp-c-neutral);--vp-button-alt-hover-bg: var(--vp-c-mute-dark);--vp-button-alt-active-border: var(--vp-c-border);--vp-button-alt-active-text: var(--vp-c-neutral);--vp-button-alt-active-bg: var(--vp-c-mute-darker);--vp-button-sponsor-border: var(--vp-c-gray-light-3);--vp-button-sponsor-text: var(--vp-c-text-light-2);--vp-button-sponsor-bg: transparent;--vp-button-sponsor-hover-border: var(--vp-c-sponsor);--vp-button-sponsor-hover-text: var(--vp-c-sponsor);--vp-button-sponsor-hover-bg: transparent;--vp-button-sponsor-active-border: var(--vp-c-sponsor);--vp-button-sponsor-active-text: var(--vp-c-sponsor);--vp-button-sponsor-active-bg: transparent}.dark{--vp-button-sponsor-border: var(--vp-c-gray-dark-1);--vp-button-sponsor-text: var(--vp-c-text-dark-2)}:root{--vp-custom-block-font-size: 14px;--vp-custom-block-code-font-size: 13px;--vp-custom-block-info-border: var(--vp-c-border);--vp-custom-block-info-text: var(--vp-c-text-2);--vp-custom-block-info-bg: var(--vp-c-bg-soft-up);--vp-custom-block-info-code-bg: var(--vp-c-bg-soft);--vp-custom-block-tip-border: var(--vp-c-green);--vp-custom-block-tip-text: var(--vp-c-green-dark);--vp-custom-block-tip-bg: var(--vp-c-bg-soft-up);--vp-custom-block-tip-code-bg: var(--vp-c-bg-soft);--vp-custom-block-warning-border: var(--vp-c-yellow);--vp-custom-block-warning-text: var(--vp-c-yellow);--vp-custom-block-warning-bg: var(--vp-c-bg-soft-up);--vp-custom-block-warning-code-bg: var(--vp-c-bg-soft);--vp-custom-block-danger-border: var(--vp-c-red);--vp-custom-block-danger-text: var(--vp-c-red);--vp-custom-block-danger-bg: var(--vp-c-bg-soft-up);--vp-custom-block-danger-code-bg: var(--vp-c-bg-soft);--vp-custom-block-details-border: var(--vp-custom-block-info-border);--vp-custom-block-details-text: var(--vp-custom-block-info-text);--vp-custom-block-details-bg: var(--vp-custom-block-info-bg);--vp-custom-block-details-code-bg: var(--vp-custom-block-details-bg)}:root{--vp-input-border-color: var(--vp-c-border);--vp-input-bg-color: var(--vp-c-bg-alt);--vp-input-hover-border-color: var(--vp-c-gray);--vp-input-switch-bg-color: var(--vp-c-mute)}:root{--vp-nav-height: 64px;--vp-nav-bg-color: var(--vp-c-bg);--vp-nav-screen-bg-color: var(--vp-c-bg)}:root{--vp-local-nav-bg-color: var(--vp-c-bg)}:root{--vp-sidebar-width: 272px;--vp-sidebar-bg-color: var(--vp-c-bg-alt)}:root{--vp-backdrop-bg-color: rgba(0, 0, 0, .6)}:root{--vp-home-hero-name-color: var(--vp-c-brand);--vp-home-hero-name-background: transparent;--vp-home-hero-image-background-image: none;--vp-home-hero-image-filter: none}:root{--vp-badge-info-border: var(--vp-c-border);--vp-badge-info-text: var(--vp-c-text-2);--vp-badge-info-bg: var(--vp-c-bg-soft-up);--vp-badge-tip-border: var(--vp-c-green-dark);--vp-badge-tip-text: var(--vp-c-green);--vp-badge-tip-bg: var(--vp-c-green-dimm-1);--vp-badge-warning-border: var(--vp-c-yellow-dark);--vp-badge-warning-text: var(--vp-c-yellow);--vp-badge-warning-bg: var(--vp-c-yellow-dimm-1);--vp-badge-danger-border: var(--vp-c-red-dark);--vp-badge-danger-text: var(--vp-c-red);--vp-badge-danger-bg: var(--vp-c-red-dimm-1)}:root{--vp-carbon-ads-text-color: var(--vp-c-text-1);--vp-carbon-ads-poweredby-color: var(--vp-c-text-2);--vp-carbon-ads-bg-color: var(--vp-c-bg-soft);--vp-carbon-ads-hover-text-color: var(--vp-c-brand);--vp-carbon-ads-hover-poweredby-color: var(--vp-c-text-1)}:root{--vp-local-search-bg: var(--vp-c-bg);--vp-local-search-result-bg: var(--vp-c-bg);--vp-local-search-result-border: var(--vp-c-divider);--vp-local-search-result-selected-bg: var(--vp-c-bg);--vp-local-search-result-selected-border: var(--vp-c-brand);--vp-local-search-highlight-bg: var(--vp-c-green-lighter);--vp-local-search-highlight-text: var(--vp-c-black)}*,:before,:after{box-sizing:border-box}html{line-height:1.4;font-size:16px;-webkit-text-size-adjust:100%}html.dark{color-scheme:dark}body{margin:0;width:100%;min-width:320px;min-height:100vh;line-height:24px;font-family:var(--vp-font-family-base);font-size:16px;font-weight:400;color:var(--vp-c-text-1);background-color:var(--vp-c-bg);direction:ltr;font-synthesis:style;text-rendering:optimizeLegibility;-webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale}h1,h2,h3,h4,h5,h6{margin:0;line-height:24px;font-size:16px;font-weight:400}p{margin:0}strong,b{font-weight:600}a,area,button,[role=button],input,label,select,summary,textarea{touch-action:manipulation}a{color:inherit;text-decoration:inherit}ol,ul{list-style:none;margin:0;padding:0}blockquote{margin:0}img,svg,video,canvas,audio,iframe,embed,object{display:block}figure{margin:0}img,video{max-width:100%;height:auto}button,input,optgroup,select,textarea{border:0;padding:0;line-height:inherit;color:inherit}button{padding:0;font-family:inherit;background-color:transparent;background-image:none}button:enabled,[role=button]:enabled{cursor:pointer}button:focus,button:focus-visible{outline:1px dotted;outline:4px auto -webkit-focus-ring-color}button:focus:not(:focus-visible){outline:none!important}input:focus,textarea:focus,select:focus{outline:none}table{border-collapse:collapse}input{background-color:transparent}input:-ms-input-placeholder,textarea:-ms-input-placeholder{color:var(--vp-c-text-3)}input::-ms-input-placeholder,textarea::-ms-input-placeholder{color:var(--vp-c-text-3)}input::placeholder,textarea::placeholder{color:var(--vp-c-text-3)}input::-webkit-outer-spin-button,input::-webkit-inner-spin-button{-webkit-appearance:none;margin:0}input[type=number]{-moz-appearance:textfield}textarea{resize:vertical}select{-webkit-appearance:none}fieldset{margin:0;padding:0}h1,h2,h3,h4,h5,h6,li,p{overflow-wrap:break-word}vite-error-overlay{z-index:9999}.visually-hidden{position:absolute;width:1px;height:1px;white-space:nowrap;clip:rect(0 0 0 0);clip-path:inset(50%);overflow:hidden}.custom-block{border:1px solid transparent;border-radius:8px;padding:16px 16px 8px;line-height:24px;font-size:var(--vp-custom-block-font-size);color:var(--vp-c-text-2)}.custom-block.info{border-color:var(--vp-custom-block-info-border);color:var(--vp-custom-block-info-text);background-color:var(--vp-custom-block-info-bg)}.custom-block.custom-block th,.custom-block.custom-block blockquote>p{font-size:var(--vp-custom-block-font-size);color:inherit}.custom-block.info code{background-color:var(--vp-custom-block-info-code-bg)}.custom-block.tip{border-color:var(--vp-custom-block-tip-border);color:var(--vp-custom-block-tip-text);background-color:var(--vp-custom-block-tip-bg)}.custom-block.tip code{background-color:var(--vp-custom-block-tip-code-bg)}.custom-block.warning{border-color:var(--vp-custom-block-warning-border);color:var(--vp-custom-block-warning-text);background-color:var(--vp-custom-block-warning-bg)}.custom-block.warning code{background-color:var(--vp-custom-block-warning-code-bg)}.custom-block.danger{border-color:var(--vp-custom-block-danger-border);color:var(--vp-custom-block-danger-text);background-color:var(--vp-custom-block-danger-bg)}.custom-block.danger code{background-color:var(--vp-custom-block-danger-code-bg)}.custom-block.details{border-color:var(--vp-custom-block-details-border);color:var(--vp-custom-block-details-text);background-color:var(--vp-custom-block-details-bg)}.custom-block.details code{background-color:var(--vp-custom-block-details-code-bg)}.custom-block-title{font-weight:600}.custom-block p+p{margin:8px 0}.custom-block.details summary{margin:0 0 8px;font-weight:700;cursor:pointer}.custom-block.details summary+p{margin:8px 0}.custom-block a{color:inherit;font-weight:600}.custom-block a:hover{text-decoration:underline}.custom-block code{font-size:var(--vp-custom-block-code-font-size)}.dark .vp-code-light{display:none}html:not(.dark) .vp-code-dark{display:none}.vp-code-group{margin-top:16px}.vp-code-group .tabs{position:relative;display:flex;margin-right:-24px;margin-left:-24px;padding:0 12px;background-color:var(--vp-code-tab-bg);overflow-x:auto;overflow-y:hidden}.vp-code-group .tabs:after{position:absolute;right:0;bottom:0;left:0;height:1px;background-color:var(--vp-code-tab-divider);content:""}@media (min-width: 640px){.vp-code-group .tabs{margin-right:0;margin-left:0;border-radius:8px 8px 0 0}}.vp-code-group .tabs input{position:absolute;opacity:0;pointer-events:none}.vp-code-group .tabs label{position:relative;display:inline-block;border-bottom:1px solid transparent;padding:0 12px;line-height:48px;font-size:14px;font-weight:500;color:var(--vp-code-tab-text-color);white-space:nowrap;cursor:pointer;transition:color .25s}.vp-code-group .tabs label:after{position:absolute;right:8px;bottom:-1px;left:8px;z-index:10;height:1px;content:"";background-color:transparent;transition:background-color .25s}.vp-code-group label:hover{color:var(--vp-code-tab-hover-text-color)}.vp-code-group input:checked+label{color:var(--vp-code-tab-active-text-color)}.vp-code-group input:checked+label:after{background-color:var(--vp-code-tab-active-bar-color)}.vp-code-group div[class*=language-]{display:none;margin-top:0!important;border-top-left-radius:0!important;border-top-right-radius:0!important}.vp-code-group div[class*=language-].active{display:block}.vp-doc h1,.vp-doc h2,.vp-doc h3,.vp-doc h4,.vp-doc h5,.vp-doc h6{position:relative;font-weight:600;outline:none}.vp-doc h1{letter-spacing:-.02em;line-height:40px;font-size:28px}.vp-doc h2{margin:48px 0 16px;border-top:1px solid var(--vp-c-divider);padding-top:24px;letter-spacing:-.02em;line-height:32px;font-size:24px}.vp-doc h3{margin:32px 0 0;letter-spacing:-.01em;line-height:28px;font-size:20px}.vp-doc .header-anchor{float:left;margin-left:-.87em;padding-right:.23em;font-weight:500;user-select:none;opacity:0;transition:color .25s,opacity .25s}.vp-doc .header-anchor:before{content:var(--vp-header-anchor-symbol)}.vp-doc h1:hover .header-anchor,.vp-doc h1 .header-anchor:focus,.vp-doc h2:hover .header-anchor,.vp-doc h2 .header-anchor:focus,.vp-doc h3:hover .header-anchor,.vp-doc h3 .header-anchor:focus,.vp-doc h4:hover .header-anchor,.vp-doc h4 .header-anchor:focus,.vp-doc h5:hover .header-anchor,.vp-doc h5 .header-anchor:focus,.vp-doc h6:hover .header-anchor,.vp-doc h6 .header-anchor:focus{opacity:1}@media (min-width: 768px){.vp-doc h1{letter-spacing:-.02em;line-height:40px;font-size:32px}}.vp-doc p,.vp-doc summary{margin:16px 0}.vp-doc p{line-height:28px}.vp-doc blockquote{margin:16px 0;border-left:2px solid var(--vp-c-divider);padding-left:16px;transition:border-color .5s}.vp-doc blockquote>p{margin:0;font-size:16px;color:var(--vp-c-text-2);transition:color .5s}.vp-doc a{font-weight:500;color:var(--vp-c-brand);text-decoration-style:dotted;transition:color .25s}.vp-doc a:hover{text-decoration:underline}.vp-doc strong{font-weight:600}.vp-doc ul,.vp-doc ol{padding-left:1.25rem;margin:16px 0}.vp-doc ul{list-style:disc}.vp-doc ol{list-style:decimal}.vp-doc li+li{margin-top:8px}.vp-doc li>ol,.vp-doc li>ul{margin:8px 0 0}.vp-doc table{display:block;border-collapse:collapse;margin:20px 0;overflow-x:auto}.vp-doc tr{border-top:1px solid var(--vp-c-divider);transition:background-color .5s}.vp-doc tr:nth-child(2n){background-color:var(--vp-c-bg-soft)}.vp-doc th,.vp-doc td{border:1px solid var(--vp-c-divider);padding:8px 16px}.vp-doc th{text-align:left;font-size:14px;font-weight:600;color:var(--vp-c-text-2);background-color:var(--vp-c-bg-soft)}.vp-doc td{font-size:14px}.vp-doc hr{margin:16px 0;border:none;border-top:1px solid var(--vp-c-divider)}.vp-doc .custom-block{margin:16px 0}.vp-doc .custom-block p{margin:8px 0;line-height:24px}.vp-doc .custom-block p:first-child{margin:0}.vp-doc .custom-block a{color:inherit;font-weight:600}.vp-doc .custom-block a:hover{text-decoration:underline}.vp-doc .custom-block code{font-size:var(--vp-custom-block-code-font-size);font-weight:700;color:inherit}.vp-doc .custom-block div[class*=language-]{margin:8px 0}.vp-doc .custom-block div[class*=language-] code{font-weight:400;background-color:transparent}.vp-doc :not(pre,h1,h2,h3,h4,h5,h6)>code{font-size:var(--vp-code-font-size)}.vp-doc :not(pre)>code{border-radius:4px;padding:3px 6px;color:var(--vp-c-text-code);background-color:var(--vp-c-mute);transition:color .5s,background-color .5s}.vp-doc h1>code,.vp-doc h2>code,.vp-doc h3>code{font-size:.9em}.vp-doc a>code{color:var(--vp-c-brand);transition:color .25s}.vp-doc a:hover>code{color:var(--vp-c-brand-dark)}.vp-doc div[class*=language-]{position:relative;margin:16px -24px;background-color:var(--vp-code-block-bg);overflow-x:auto;transition:background-color .5s}@media (min-width: 640px){.vp-doc div[class*=language-]{border-radius:8px;margin:16px 0}}@media (max-width: 639px){.vp-doc li div[class*=language-]{border-radius:8px 0 0 8px}}.vp-doc div[class*=language-]+div[class*=language-],.vp-doc div[class$=-api]+div[class*=language-],.vp-doc div[class*=language-]+div[class$=-api]>div[class*=language-]{margin-top:-8px}.vp-doc [class*=language-] pre,.vp-doc [class*=language-] code{direction:ltr;text-align:left;white-space:pre;word-spacing:normal;word-break:normal;word-wrap:normal;-moz-tab-size:4;-o-tab-size:4;tab-size:4;-webkit-hyphens:none;-moz-hyphens:none;-ms-hyphens:none;hyphens:none}.vp-doc [class*=language-] pre{position:relative;z-index:1;margin:0;padding:20px 0;background:transparent;overflow-x:auto}.vp-doc [class*=language-] code{display:block;padding:0 24px;width:fit-content;min-width:100%;line-height:var(--vp-code-line-height);font-size:var(--vp-code-font-size);color:var(--vp-code-block-color);transition:color .5s}.vp-doc [class*=language-] code .highlighted{background-color:var(--vp-code-line-highlight-color);transition:background-color .5s;margin:0 -24px;padding:0 24px;width:calc(100% + 48px);display:inline-block}.vp-doc [class*=language-] code .highlighted.error{background-color:var(--vp-code-line-error-color)}.vp-doc [class*=language-] code .highlighted.warning{background-color:var(--vp-code-line-warning-color)}.vp-doc [class*=language-] code .diff{transition:background-color .5s;margin:0 -24px;padding:0 24px;width:calc(100% + 48px);display:inline-block}.vp-doc [class*=language-] code .diff:before{position:absolute;left:10px}.vp-doc [class*=language-] .has-focused-lines .line:not(.has-focus){filter:blur(.095rem);opacity:.4;transition:filter .35s,opacity .35s}.vp-doc [class*=language-] .has-focused-lines .line:not(.has-focus){opacity:.7;transition:filter .35s,opacity .35s}.vp-doc [class*=language-]:hover .has-focused-lines .line:not(.has-focus){filter:blur(0);opacity:1}.vp-doc [class*=language-] code .diff.remove{background-color:var(--vp-code-line-diff-remove-color);opacity:.7}.vp-doc [class*=language-] code .diff.remove:before{content:"-";color:var(--vp-code-line-diff-remove-symbol-color)}.vp-doc [class*=language-] code .diff.add{background-color:var(--vp-code-line-diff-add-color)}.vp-doc [class*=language-] code .diff.add:before{content:"+";color:var(--vp-code-line-diff-add-symbol-color)}.vp-doc div[class*=language-].line-numbers-mode{padding-left:32px}.vp-doc .line-numbers-wrapper{position:absolute;top:0;bottom:0;left:0;z-index:3;border-right:1px solid var(--vp-code-block-divider-color);padding-top:20px;width:32px;text-align:center;font-family:var(--vp-font-family-mono);line-height:var(--vp-code-line-height);font-size:var(--vp-code-font-size);color:var(--vp-code-line-number-color);transition:border-color .5s,color .5s}.vp-doc [class*=language-]>button.copy{direction:ltr;position:absolute;top:12px;right:12px;z-index:3;border:1px solid var(--vp-code-copy-code-border-color);border-radius:4px;width:40px;height:40px;background-color:var(--vp-code-copy-code-bg);opacity:0;cursor:pointer;background-image:var(--vp-icon-copy);background-position:50%;background-size:20px;background-repeat:no-repeat;transition:border-color .25s,background-color .25s,opacity .25s}.vp-doc [class*=language-]:hover>button.copy,.vp-doc [class*=language-]>button.copy:focus{opacity:1}.vp-doc [class*=language-]>button.copy:hover,.vp-doc [class*=language-]>button.copy.copied{border-color:var(--vp-code-copy-code-hover-border-color);background-color:var(--vp-code-copy-code-hover-bg)}.vp-doc [class*=language-]>button.copy.copied,.vp-doc [class*=language-]>button.copy:hover.copied{border-radius:0 4px 4px 0;background-color:var(--vp-code-copy-code-hover-bg);background-image:var(--vp-icon-copied)}.vp-doc [class*=language-]>button.copy.copied:before,.vp-doc [class*=language-]>button.copy:hover.copied:before{position:relative;top:-1px;left:-65px;display:flex;justify-content:center;align-items:center;border:1px solid var(--vp-code-copy-code-hover-border-color);border-right:0;border-radius:4px 0 0 4px;width:64px;height:40px;text-align:center;font-size:12px;font-weight:500;color:var(--vp-code-copy-code-active-text);background-color:var(--vp-code-copy-code-hover-bg);white-space:nowrap;content:"Copied"}.vp-doc [class*=language-]>span.lang{position:absolute;top:2px;right:8px;z-index:2;font-size:12px;font-weight:500;color:var(--vp-c-code-dimm);transition:color .4s,opacity .4s}.vp-doc [class*=language-]:hover>button.copy+span.lang,.vp-doc [class*=language-]>button.copy:focus+span.lang{opacity:0}.vp-doc .VPTeamMembers{margin-top:24px}.vp-doc .VPTeamMembers.small.count-1 .container{margin:0!important;max-width:calc((100% - 24px)/2)!important}.vp-doc .VPTeamMembers.small.count-2 .container,.vp-doc .VPTeamMembers.small.count-3 .container{max-width:100%!important}.vp-doc .VPTeamMembers.medium.count-1 .container{margin:0!important;max-width:calc((100% - 24px)/2)!important}.vp-sponsor{border-radius:16px;overflow:hidden}.vp-sponsor.aside{border-radius:12px}.vp-sponsor-section+.vp-sponsor-section{margin-top:4px}.vp-sponsor-tier{margin-bottom:4px;text-align:center;letter-spacing:1px;line-height:24px;width:100%;font-weight:600;color:var(--vp-c-text-2);background-color:var(--vp-c-bg-soft)}.vp-sponsor.normal .vp-sponsor-tier{padding:13px 0 11px;font-size:14px}.vp-sponsor.aside .vp-sponsor-tier{padding:9px 0 7px;font-size:12px}.vp-sponsor-grid+.vp-sponsor-tier{margin-top:4px}.vp-sponsor-grid{display:flex;flex-wrap:wrap;gap:4px}.vp-sponsor-grid.xmini .vp-sponsor-grid-link{height:64px}.vp-sponsor-grid.xmini .vp-sponsor-grid-image{max-width:64px;max-height:22px}.vp-sponsor-grid.mini .vp-sponsor-grid-link{height:72px}.vp-sponsor-grid.mini .vp-sponsor-grid-image{max-width:96px;max-height:24px}.vp-sponsor-grid.small .vp-sponsor-grid-link{height:96px}.vp-sponsor-grid.small .vp-sponsor-grid-image{max-width:96px;max-height:24px}.vp-sponsor-grid.medium .vp-sponsor-grid-link{height:112px}.vp-sponsor-grid.medium .vp-sponsor-grid-image{max-width:120px;max-height:36px}.vp-sponsor-grid.big .vp-sponsor-grid-link{height:184px}.vp-sponsor-grid.big .vp-sponsor-grid-image{max-width:192px;max-height:56px}.vp-sponsor-grid[data-vp-grid="2"] .vp-sponsor-grid-item{width:calc((100% - 4px)/2)}.vp-sponsor-grid[data-vp-grid="3"] .vp-sponsor-grid-item{width:calc((100% - 4px * 2) / 3)}.vp-sponsor-grid[data-vp-grid="4"] .vp-sponsor-grid-item{width:calc((100% - 12px)/4)}.vp-sponsor-grid[data-vp-grid="5"] .vp-sponsor-grid-item{width:calc((100% - 16px)/5)}.vp-sponsor-grid[data-vp-grid="6"] .vp-sponsor-grid-item{width:calc((100% - 4px * 5) / 6)}.vp-sponsor-grid-item{flex-shrink:0;width:100%;background-color:var(--vp-c-bg-soft);transition:background-color .25s}.vp-sponsor-grid-item:hover{background-color:var(--vp-c-bg-soft-down)}.vp-sponsor-grid-item:hover .vp-sponsor-grid-image{filter:grayscale(0) invert(0)}.vp-sponsor-grid-item.empty:hover{background-color:var(--vp-c-bg-soft)}.dark .vp-sponsor-grid-item:hover{background-color:var(--vp-c-white)}.dark .vp-sponsor-grid-item.empty:hover{background-color:var(--vp-c-bg-soft)}.vp-sponsor-grid-link{display:flex}.vp-sponsor-grid-box{display:flex;justify-content:center;align-items:center;width:100%}.vp-sponsor-grid-image{max-width:100%;filter:grayscale(1);transition:filter .25s}.dark .vp-sponsor-grid-image{filter:grayscale(1) invert(1)}.VPBadge[data-v-b8591dbf]{display:inline-block;margin-left:2px;border:1px solid transparent;border-radius:10px;padding:0 8px;line-height:18px;font-size:12px;font-weight:600;transform:translateY(-2px)}h1 .VPBadge[data-v-b8591dbf],h2 .VPBadge[data-v-b8591dbf],h3 .VPBadge[data-v-b8591dbf],h4 .VPBadge[data-v-b8591dbf],h5 .VPBadge[data-v-b8591dbf],h6 .VPBadge[data-v-b8591dbf]{vertical-align:top}h2 .VPBadge[data-v-b8591dbf]{border-radius:11px;line-height:20px}.VPBadge.info[data-v-b8591dbf]{border-color:var(--vp-badge-info-border);color:var(--vp-badge-info-text);background-color:var(--vp-badge-info-bg)}.VPBadge.tip[data-v-b8591dbf]{border-color:var(--vp-badge-tip-border);color:var(--vp-badge-tip-text);background-color:var(--vp-badge-tip-bg)}.VPBadge.warning[data-v-b8591dbf]{border-color:var(--vp-badge-warning-border);color:var(--vp-badge-warning-text);background-color:var(--vp-badge-warning-bg)}.VPBadge.danger[data-v-b8591dbf]{border-color:var(--vp-badge-danger-border);color:var(--vp-badge-danger-text);background-color:var(--vp-badge-danger-bg)}.VPSkipLink[data-v-aa63c34f]{top:8px;left:8px;padding:8px 16px;z-index:999;border-radius:8px;font-size:12px;font-weight:700;text-decoration:none;color:var(--vp-c-brand);box-shadow:var(--vp-shadow-3);background-color:var(--vp-c-bg)}.VPSkipLink[data-v-aa63c34f]:focus{height:auto;width:auto;clip:auto;clip-path:none}.dark .VPSkipLink[data-v-aa63c34f]{color:var(--vp-c-green)}@media (min-width: 1280px){.VPSkipLink[data-v-aa63c34f]{top:14px;left:16px}}.VPBackdrop[data-v-d0d4b99a]{position:fixed;top:0;right:0;bottom:0;left:0;z-index:var(--vp-z-index-backdrop);background:var(--vp-backdrop-bg-color);transition:opacity .5s}.VPBackdrop.fade-enter-from[data-v-d0d4b99a],.VPBackdrop.fade-leave-to[data-v-d0d4b99a]{opacity:0}.VPBackdrop.fade-leave-active[data-v-d0d4b99a]{transition-duration:.25s}@media (min-width: 1280px){.VPBackdrop[data-v-d0d4b99a]{display:none}}html:not(.dark) .VPImage.dark[data-v-6f75ecd0]{display:none}.dark .VPImage.light[data-v-6f75ecd0]{display:none}.title[data-v-75402b34]{display:flex;align-items:center;border-bottom:1px solid transparent;width:100%;height:var(--vp-nav-height);font-size:16px;font-weight:600;color:var(--vp-c-text-1);transition:opacity .25s}.title[data-v-75402b34]:hover{opacity:.6}@media (min-width: 960px){.title[data-v-75402b34]{flex-shrink:0}.VPNavBarTitle.has-sidebar .title[data-v-75402b34]{border-bottom-color:var(--vp-c-divider)}}[data-v-75402b34] .logo{margin-right:8px;height:24px}/*! @docsearch/css 3.3.4 | MIT License | © Algolia, Inc. and contributors | https://docsearch.algolia.com */:root{--docsearch-primary-color:#5468ff;--docsearch-text-color:#1c1e21;--docsearch-spacing:12px;--docsearch-icon-stroke-width:1.4;--docsearch-highlight-color:var(--docsearch-primary-color);--docsearch-muted-color:#969faf;--docsearch-container-background:rgba(101,108,133,.8);--docsearch-logo-color:#5468ff;--docsearch-modal-width:560px;--docsearch-modal-height:600px;--docsearch-modal-background:#f5f6f7;--docsearch-modal-shadow:inset 1px 1px 0 0 hsla(0,0%,100%,.5),0 3px 8px 0 #555a64;--docsearch-searchbox-height:56px;--docsearch-searchbox-background:#ebedf0;--docsearch-searchbox-focus-background:#fff;--docsearch-searchbox-shadow:inset 0 0 0 2px var(--docsearch-primary-color);--docsearch-hit-height:56px;--docsearch-hit-color:#444950;--docsearch-hit-active-color:#fff;--docsearch-hit-background:#fff;--docsearch-hit-shadow:0 1px 3px 0 #d4d9e1;--docsearch-key-gradient:linear-gradient(-225deg,#d5dbe4,#f8f8f8);--docsearch-key-shadow:inset 0 -2px 0 0 #cdcde6,inset 0 0 1px 1px #fff,0 1px 2px 1px rgba(30,35,90,.4);--docsearch-footer-height:44px;--docsearch-footer-background:#fff;--docsearch-footer-shadow:0 -1px 0 0 #e0e3e8,0 -3px 6px 0 rgba(69,98,155,.12)}html[data-theme=dark]{--docsearch-text-color:#f5f6f7;--docsearch-container-background:rgba(9,10,17,.8);--docsearch-modal-background:#15172a;--docsearch-modal-shadow:inset 1px 1px 0 0 #2c2e40,0 3px 8px 0 #000309;--docsearch-searchbox-background:#090a11;--docsearch-searchbox-focus-background:#000;--docsearch-hit-color:#bec3c9;--docsearch-hit-shadow:none;--docsearch-hit-background:#090a11;--docsearch-key-gradient:linear-gradient(-26.5deg,#565872,#31355b);--docsearch-key-shadow:inset 0 -2px 0 0 #282d55,inset 0 0 1px 1px #51577d,0 2px 2px 0 rgba(3,4,9,.3);--docsearch-footer-background:#1e2136;--docsearch-footer-shadow:inset 0 1px 0 0 rgba(73,76,106,.5),0 -4px 8px 0 rgba(0,0,0,.2);--docsearch-logo-color:#fff;--docsearch-muted-color:#7f8497}.DocSearch-Button{align-items:center;background:var(--docsearch-searchbox-background);border:0;border-radius:40px;color:var(--docsearch-muted-color);cursor:pointer;display:flex;font-weight:500;height:36px;justify-content:space-between;margin:0 0 0 16px;padding:0 8px;user-select:none}.DocSearch-Button:active,.DocSearch-Button:focus,.DocSearch-Button:hover{background:var(--docsearch-searchbox-focus-background);box-shadow:var(--docsearch-searchbox-shadow);color:var(--docsearch-text-color);outline:none}.DocSearch-Button-Container{align-items:center;display:flex}.DocSearch-Search-Icon{stroke-width:1.6}.DocSearch-Button .DocSearch-Search-Icon{color:var(--docsearch-text-color)}.DocSearch-Button-Placeholder{font-size:1rem;padding:0 12px 0 6px}.DocSearch-Button-Keys{display:flex;min-width:calc(40px + .8em)}.DocSearch-Button-Key{align-items:center;background:var(--docsearch-key-gradient);border-radius:3px;box-shadow:var(--docsearch-key-shadow);color:var(--docsearch-muted-color);display:flex;height:18px;justify-content:center;margin-right:.4em;position:relative;padding:0 0 2px;border:0;top:-1px;width:20px}@media (max-width:768px){.DocSearch-Button-Keys,.DocSearch-Button-Placeholder{display:none}}.DocSearch--active{overflow:hidden!important}.DocSearch-Container,.DocSearch-Container *{box-sizing:border-box}.DocSearch-Container{background-color:var(--docsearch-container-background);height:100vh;left:0;position:fixed;top:0;width:100vw;z-index:200}.DocSearch-Container a{text-decoration:none}.DocSearch-Link{appearance:none;background:none;border:0;color:var(--docsearch-highlight-color);cursor:pointer;font:inherit;margin:0;padding:0}.DocSearch-Modal{background:var(--docsearch-modal-background);border-radius:6px;box-shadow:var(--docsearch-modal-shadow);flex-direction:column;margin:60px auto auto;max-width:var(--docsearch-modal-width);position:relative}.DocSearch-SearchBar{display:flex;padding:var(--docsearch-spacing) var(--docsearch-spacing) 0}.DocSearch-Form{align-items:center;background:var(--docsearch-searchbox-focus-background);border-radius:4px;box-shadow:var(--docsearch-searchbox-shadow);display:flex;height:var(--docsearch-searchbox-height);margin:0;padding:0 var(--docsearch-spacing);position:relative;width:100%}.DocSearch-Input{appearance:none;background:transparent;border:0;color:var(--docsearch-text-color);flex:1;font:inherit;font-size:1.2em;height:100%;outline:none;padding:0 0 0 8px;width:80%}.DocSearch-Input::placeholder{color:var(--docsearch-muted-color);opacity:1}.DocSearch-Input::-webkit-search-cancel-button,.DocSearch-Input::-webkit-search-decoration,.DocSearch-Input::-webkit-search-results-button,.DocSearch-Input::-webkit-search-results-decoration{display:none}.DocSearch-LoadingIndicator,.DocSearch-MagnifierLabel,.DocSearch-Reset{margin:0;padding:0}.DocSearch-MagnifierLabel,.DocSearch-Reset{align-items:center;color:var(--docsearch-highlight-color);display:flex;justify-content:center}.DocSearch-Container--Stalled .DocSearch-MagnifierLabel,.DocSearch-LoadingIndicator{display:none}.DocSearch-Container--Stalled .DocSearch-LoadingIndicator{align-items:center;color:var(--docsearch-highlight-color);display:flex;justify-content:center}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Reset{animation:none;appearance:none;background:none;border:0;border-radius:50%;color:var(--docsearch-icon-color);cursor:pointer;right:0;stroke-width:var(--docsearch-icon-stroke-width)}}.DocSearch-Reset{animation:fade-in .1s ease-in forwards;appearance:none;background:none;border:0;border-radius:50%;color:var(--docsearch-icon-color);cursor:pointer;padding:2px;right:0;stroke-width:var(--docsearch-icon-stroke-width)}.DocSearch-Reset[hidden]{display:none}.DocSearch-Reset:hover{color:var(--docsearch-highlight-color)}.DocSearch-LoadingIndicator svg,.DocSearch-MagnifierLabel svg{height:24px;width:24px}.DocSearch-Cancel{display:none}.DocSearch-Dropdown{max-height:calc(var(--docsearch-modal-height) - var(--docsearch-searchbox-height) - var(--docsearch-spacing) - var(--docsearch-footer-height));min-height:var(--docsearch-spacing);overflow-y:auto;overflow-y:overlay;padding:0 var(--docsearch-spacing);scrollbar-color:var(--docsearch-muted-color) var(--docsearch-modal-background);scrollbar-width:thin}.DocSearch-Dropdown::-webkit-scrollbar{width:12px}.DocSearch-Dropdown::-webkit-scrollbar-track{background:transparent}.DocSearch-Dropdown::-webkit-scrollbar-thumb{background-color:var(--docsearch-muted-color);border:3px solid var(--docsearch-modal-background);border-radius:20px}.DocSearch-Dropdown ul{list-style:none;margin:0;padding:0}.DocSearch-Label{font-size:.75em;line-height:1.6em}.DocSearch-Help,.DocSearch-Label{color:var(--docsearch-muted-color)}.DocSearch-Help{font-size:.9em;margin:0;user-select:none}.DocSearch-Title{font-size:1.2em}.DocSearch-Logo a{display:flex}.DocSearch-Logo svg{color:var(--docsearch-logo-color);margin-left:8px}.DocSearch-Hits:last-of-type{margin-bottom:24px}.DocSearch-Hits mark{background:none;color:var(--docsearch-highlight-color)}.DocSearch-HitsFooter{color:var(--docsearch-muted-color);display:flex;font-size:.85em;justify-content:center;margin-bottom:var(--docsearch-spacing);padding:var(--docsearch-spacing)}.DocSearch-HitsFooter a{border-bottom:1px solid;color:inherit}.DocSearch-Hit{border-radius:4px;display:flex;padding-bottom:4px;position:relative}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit--deleting{transition:none}}.DocSearch-Hit--deleting{opacity:0;transition:all .25s linear}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit--favoriting{transition:none}}.DocSearch-Hit--favoriting{transform:scale(0);transform-origin:top center;transition:all .25s linear;transition-delay:.25s}.DocSearch-Hit a{background:var(--docsearch-hit-background);border-radius:4px;box-shadow:var(--docsearch-hit-shadow);display:block;padding-left:var(--docsearch-spacing);width:100%}.DocSearch-Hit-source{background:var(--docsearch-modal-background);color:var(--docsearch-highlight-color);font-size:.85em;font-weight:600;line-height:32px;margin:0 -4px;padding:8px 4px 0;position:sticky;top:0;z-index:10}.DocSearch-Hit-Tree{color:var(--docsearch-muted-color);height:var(--docsearch-hit-height);opacity:.5;stroke-width:var(--docsearch-icon-stroke-width);width:24px}.DocSearch-Hit[aria-selected=true] a{background-color:var(--docsearch-highlight-color)}.DocSearch-Hit[aria-selected=true] mark{text-decoration:underline}.DocSearch-Hit-Container{align-items:center;color:var(--docsearch-hit-color);display:flex;flex-direction:row;height:var(--docsearch-hit-height);padding:0 var(--docsearch-spacing) 0 0}.DocSearch-Hit-icon{height:20px;width:20px}.DocSearch-Hit-action,.DocSearch-Hit-icon{color:var(--docsearch-muted-color);stroke-width:var(--docsearch-icon-stroke-width)}.DocSearch-Hit-action{align-items:center;display:flex;height:22px;width:22px}.DocSearch-Hit-action svg{display:block;height:18px;width:18px}.DocSearch-Hit-action+.DocSearch-Hit-action{margin-left:6px}.DocSearch-Hit-action-button{appearance:none;background:none;border:0;border-radius:50%;color:inherit;cursor:pointer;padding:2px}svg.DocSearch-Hit-Select-Icon{display:none}.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-Select-Icon{display:block}.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{background:rgba(0,0,0,.2);transition:background-color .1s ease-in}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{transition:none}}.DocSearch-Hit-action-button:focus path,.DocSearch-Hit-action-button:hover path{fill:#fff}.DocSearch-Hit-content-wrapper{display:flex;flex:1 1 auto;flex-direction:column;font-weight:500;justify-content:center;line-height:1.2em;margin:0 8px;overflow-x:hidden;position:relative;text-overflow:ellipsis;white-space:nowrap;width:80%}.DocSearch-Hit-title{font-size:.9em}.DocSearch-Hit-path{color:var(--docsearch-muted-color);font-size:.75em}.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-action,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-icon,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-path,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-text,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-title,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-Tree,.DocSearch-Hit[aria-selected=true] mark{color:var(--docsearch-hit-active-color)!important}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{background:rgba(0,0,0,.2);transition:none}}.DocSearch-ErrorScreen,.DocSearch-NoResults,.DocSearch-StartScreen{font-size:.9em;margin:0 auto;padding:36px 0;text-align:center;width:80%}.DocSearch-Screen-Icon{color:var(--docsearch-muted-color);padding-bottom:12px}.DocSearch-NoResults-Prefill-List{display:inline-block;padding-bottom:24px;text-align:left}.DocSearch-NoResults-Prefill-List ul{display:inline-block;padding:8px 0 0}.DocSearch-NoResults-Prefill-List li{list-style-position:inside;list-style-type:"» "}.DocSearch-Prefill{appearance:none;background:none;border:0;border-radius:1em;color:var(--docsearch-highlight-color);cursor:pointer;display:inline-block;font-size:1em;font-weight:700;padding:0}.DocSearch-Prefill:focus,.DocSearch-Prefill:hover{outline:none;text-decoration:underline}.DocSearch-Footer{align-items:center;background:var(--docsearch-footer-background);border-radius:0 0 8px 8px;box-shadow:var(--docsearch-footer-shadow);display:flex;flex-direction:row-reverse;flex-shrink:0;height:var(--docsearch-footer-height);justify-content:space-between;padding:0 var(--docsearch-spacing);position:relative;user-select:none;width:100%;z-index:300}.DocSearch-Commands{color:var(--docsearch-muted-color);display:flex;list-style:none;margin:0;padding:0}.DocSearch-Commands li{align-items:center;display:flex}.DocSearch-Commands li:not(:last-of-type){margin-right:.8em}.DocSearch-Commands-Key{align-items:center;background:var(--docsearch-key-gradient);border-radius:2px;box-shadow:var(--docsearch-key-shadow);display:flex;height:18px;justify-content:center;margin-right:.4em;padding:0 0 1px;color:var(--docsearch-muted-color);border:0;width:20px}@media (max-width:768px){:root{--docsearch-spacing:10px;--docsearch-footer-height:40px}.DocSearch-Dropdown{height:100%}.DocSearch-Container{height:100vh;height:-webkit-fill-available;height:calc(var(--docsearch-vh, 1vh)*100);position:absolute}.DocSearch-Footer{border-radius:0;bottom:0;position:absolute}.DocSearch-Hit-content-wrapper{display:flex;position:relative;width:80%}.DocSearch-Modal{border-radius:0;box-shadow:none;height:100vh;height:-webkit-fill-available;height:calc(var(--docsearch-vh, 1vh)*100);margin:0;max-width:100%;width:100%}.DocSearch-Dropdown{max-height:calc(var(--docsearch-vh, 1vh)*100 - var(--docsearch-searchbox-height) - var(--docsearch-spacing) - var(--docsearch-footer-height))}.DocSearch-Cancel{appearance:none;background:none;border:0;color:var(--docsearch-highlight-color);cursor:pointer;display:inline-block;flex:none;font:inherit;font-size:1em;font-weight:500;margin-left:var(--docsearch-spacing);outline:none;overflow:hidden;padding:0;user-select:none;white-space:nowrap}.DocSearch-Commands,.DocSearch-Hit-Tree{display:none}}@keyframes fade-in{0%{opacity:0}to{opacity:1}}.DocSearch{--docsearch-primary-color: var(--vp-c-brand);--docsearch-highlight-color: var(--docsearch-primary-color);--docsearch-text-color: var(--vp-c-text-1);--docsearch-muted-color: var(--vp-c-text-2);--docsearch-searchbox-shadow: none;--docsearch-searchbox-focus-background: transparent;--docsearch-key-gradient: transparent;--docsearch-key-shadow: none;--docsearch-modal-background: var(--vp-c-bg-soft);--docsearch-footer-background: var(--vp-c-bg)}.dark .DocSearch{--docsearch-modal-shadow: none;--docsearch-footer-shadow: none;--docsearch-logo-color: var(--vp-c-text-2);--docsearch-hit-background: var(--vp-c-bg-soft-mute);--docsearch-hit-color: var(--vp-c-text-2);--docsearch-hit-shadow: none}.DocSearch-Button{display:flex;justify-content:center;align-items:center;margin:0;padding:0;width:32px;height:55px;background:transparent;transition:border-color .25s}.DocSearch-Button:hover{background:transparent}.DocSearch-Button:focus{outline:1px dotted;outline:5px auto -webkit-focus-ring-color}.DocSearch-Button:focus:not(:focus-visible){outline:none!important}@media (min-width: 768px){.DocSearch-Button{justify-content:flex-start;border:1px solid transparent;border-radius:8px;padding:0 10px 0 12px;width:100%;height:40px;background-color:var(--vp-c-bg-alt)}.DocSearch-Button:hover{border-color:var(--vp-c-brand);background:var(--vp-c-bg-alt)}}.DocSearch-Button .DocSearch-Button-Container{display:flex;align-items:center}.DocSearch-Button .DocSearch-Search-Icon{position:relative;width:16px;height:16px;color:var(--vp-c-text-1);fill:currentColor;transition:color .5s}.DocSearch-Button:hover .DocSearch-Search-Icon{color:var(--vp-c-text-1)}@media (min-width: 768px){.DocSearch-Button .DocSearch-Search-Icon{top:1px;margin-right:8px;width:14px;height:14px;color:var(--vp-c-text-2)}}.DocSearch-Button .DocSearch-Button-Placeholder{display:none;margin-top:2px;padding:0 16px 0 0;font-size:13px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.DocSearch-Button:hover .DocSearch-Button-Placeholder{color:var(--vp-c-text-1)}@media (min-width: 768px){.DocSearch-Button .DocSearch-Button-Placeholder{display:inline-block}}.DocSearch-Button .DocSearch-Button-Keys{direction:ltr;display:none;min-width:auto}@media (min-width: 768px){.DocSearch-Button .DocSearch-Button-Keys{display:flex;align-items:center}}.DocSearch-Button .DocSearch-Button-Key{display:block;margin:2px 0 0;border:1px solid var(--vp-c-divider);border-right:none;border-radius:4px 0 0 4px;padding-left:6px;min-width:0;width:auto;height:22px;line-height:22px;font-family:var(--vp-font-family-base);font-size:12px;font-weight:500;transition:color .5s,border-color .5s}.DocSearch-Button .DocSearch-Button-Key+.DocSearch-Button-Key{border-right:1px solid var(--vp-c-divider);border-left:none;border-radius:0 4px 4px 0;padding-left:2px;padding-right:6px}.DocSearch-Button .DocSearch-Button-Key:first-child{font-size:1px;letter-spacing:-12px;color:transparent}.DocSearch-Button .DocSearch-Button-Key:first-child:after{content:var(--vp-meta-key);font-size:12px;letter-spacing:normal;color:var(--docsearch-muted-color)}.DocSearch-Button .DocSearch-Button-Key:first-child>*{display:none}.VPNavBarSearch{display:flex;align-items:center}@media (min-width: 768px){.VPNavBarSearch{flex-grow:1;padding-left:24px}}@media (min-width: 960px){.VPNavBarSearch{padding-left:32px}}.dark .DocSearch-Footer{border-top:1px solid var(--vp-c-divider)}.DocSearch-Form{border:1px solid var(--vp-c-brand);background-color:var(--vp-c-white)}.dark .DocSearch-Form{background-color:var(--vp-c-bg-soft-mute)}.DocSearch-Screen-Icon>svg{margin:auto}.icon[data-v-a72b6228]{display:inline-block;margin-top:-1px;margin-left:4px;width:11px;height:11px;fill:var(--vp-c-text-3);transition:fill .25s;flex-shrink:0}.VPNavBarMenuLink[data-v-fa5c4352]{display:flex;align-items:center;padding:0 12px;line-height:var(--vp-nav-height);font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.VPNavBarMenuLink.active[data-v-fa5c4352],.VPNavBarMenuLink[data-v-fa5c4352]:hover{color:var(--vp-c-brand)}.VPMenuGroup+.VPMenuLink[data-v-dcf4ab55]{margin:12px -12px 0;border-top:1px solid var(--vp-c-divider);padding:12px 12px 0}.link[data-v-dcf4ab55]{display:block;border-radius:6px;padding:0 12px;line-height:32px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);white-space:nowrap;transition:background-color .25s,color .25s}.link[data-v-dcf4ab55]:hover{color:var(--vp-c-brand);background-color:var(--vp-c-bg-elv-mute)}.link.active[data-v-dcf4ab55]{color:var(--vp-c-brand)}.VPMenuGroup[data-v-b48b6f40]{margin:12px -12px 0;border-top:1px solid var(--vp-c-divider);padding:12px 12px 0}.VPMenuGroup[data-v-b48b6f40]:first-child{margin-top:0;border-top:0;padding-top:0}.VPMenuGroup+.VPMenuGroup[data-v-b48b6f40]{margin-top:12px;border-top:1px solid var(--vp-c-divider)}.title[data-v-b48b6f40]{padding:0 12px;line-height:32px;font-size:14px;font-weight:600;color:var(--vp-c-text-2);white-space:nowrap;transition:color .25s}.VPMenu[data-v-ed414532]{border-radius:12px;padding:12px;min-width:128px;border:1px solid var(--vp-c-divider);background-color:var(--vp-c-bg-elv);box-shadow:var(--vp-shadow-3);transition:background-color .5s;max-height:calc(100vh - var(--vp-nav-height));overflow-y:auto}.VPMenu[data-v-ed414532] .group{margin:0 -12px;padding:0 12px 12px}.VPMenu[data-v-ed414532] .group+.group{border-top:1px solid var(--vp-c-divider);padding:11px 12px 12px}.VPMenu[data-v-ed414532] .group:last-child{padding-bottom:0}.VPMenu[data-v-ed414532] .group+.item{border-top:1px solid var(--vp-c-divider);padding:11px 16px 0}.VPMenu[data-v-ed414532] .item{padding:0 16px;white-space:nowrap}.VPMenu[data-v-ed414532] .label{flex-grow:1;line-height:28px;font-size:12px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.VPMenu[data-v-ed414532] .action{padding-left:24px}.VPFlyout[data-v-d146844d]{position:relative}.VPFlyout[data-v-d146844d]:hover{color:var(--vp-c-brand);transition:color .25s}.VPFlyout:hover .text[data-v-d146844d]{color:var(--vp-c-text-2)}.VPFlyout:hover .icon[data-v-d146844d]{fill:var(--vp-c-text-2)}.VPFlyout.active .text[data-v-d146844d]{color:var(--vp-c-brand)}.VPFlyout.active:hover .text[data-v-d146844d]{color:var(--vp-c-brand-dark)}.VPFlyout:hover .menu[data-v-d146844d],.button[aria-expanded=true]+.menu[data-v-d146844d]{opacity:1;visibility:visible;transform:translateY(0)}.button[data-v-d146844d]{display:flex;align-items:center;padding:0 12px;height:var(--vp-nav-height);color:var(--vp-c-text-1);transition:color .5s}.text[data-v-d146844d]{display:flex;align-items:center;line-height:var(--vp-nav-height);font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.option-icon[data-v-d146844d]{margin-right:0;width:16px;height:16px;fill:currentColor}.text-icon[data-v-d146844d]{margin-left:4px;width:14px;height:14px;fill:currentColor}.icon[data-v-d146844d]{width:20px;height:20px;fill:currentColor;transition:fill .25s}.menu[data-v-d146844d]{position:absolute;top:calc(var(--vp-nav-height) / 2 + 20px);right:0;opacity:0;visibility:hidden;transition:opacity .25s,visibility .25s,transform .25s}.VPNavBarMenu[data-v-a0e7a303]{display:none}@media (min-width: 768px){.VPNavBarMenu[data-v-a0e7a303]{display:flex}}.VPNavBarTranslations[data-v-9047dbea]{display:none}@media (min-width: 1280px){.VPNavBarTranslations[data-v-9047dbea]{display:flex;align-items:center}}.title[data-v-9047dbea]{padding:0 24px 0 12px;line-height:32px;font-size:14px;font-weight:700;color:var(--vp-c-text-1)}.VPSwitch[data-v-fb9fb2bc]{position:relative;border-radius:11px;display:block;width:40px;height:22px;flex-shrink:0;border:1px solid var(--vp-input-border-color);background-color:var(--vp-input-switch-bg-color);transition:border-color .25s}.VPSwitch[data-v-fb9fb2bc]:hover{border-color:var(--vp-input-hover-border-color)}.check[data-v-fb9fb2bc]{position:absolute;top:1px;left:1px;width:18px;height:18px;border-radius:50%;background-color:var(--vp-c-neutral-inverse);box-shadow:var(--vp-shadow-1);transition:transform .25s}.icon[data-v-fb9fb2bc]{position:relative;display:block;width:18px;height:18px;border-radius:50%;overflow:hidden}.icon[data-v-fb9fb2bc] svg{position:absolute;top:3px;left:3px;width:12px;height:12px;fill:var(--vp-c-text-2)}.dark .icon[data-v-fb9fb2bc] svg{fill:var(--vp-c-text-1);transition:opacity .25s}.sun[data-v-5314f25c]{opacity:1}.moon[data-v-5314f25c],.dark .sun[data-v-5314f25c]{opacity:0}.dark .moon[data-v-5314f25c]{opacity:1}.dark .VPSwitchAppearance[data-v-5314f25c] .check{transform:translate(18px)}.VPNavBarAppearance[data-v-ba9af2dd]{display:none}@media (min-width: 1280px){.VPNavBarAppearance[data-v-ba9af2dd]{display:flex;align-items:center}}.VPSocialLink[data-v-f4bd6624]{display:flex;justify-content:center;align-items:center;width:36px;height:36px;color:var(--vp-c-text-2);transition:color .5s}.VPSocialLink[data-v-f4bd6624]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPSocialLink[data-v-f4bd6624]>svg{width:20px;height:20px;fill:currentColor}.VPSocialLinks[data-v-fea619a9]{display:flex;flex-wrap:wrap;justify-content:center}.VPNavBarSocialLinks[data-v-49754fc9]{display:none}@media (min-width: 1280px){.VPNavBarSocialLinks[data-v-49754fc9]{display:flex;align-items:center}}.VPNavBarExtra[data-v-5ac0399f]{display:none;margin-right:-12px}@media (min-width: 768px){.VPNavBarExtra[data-v-5ac0399f]{display:block}}@media (min-width: 1280px){.VPNavBarExtra[data-v-5ac0399f]{display:none}}.trans-title[data-v-5ac0399f]{padding:0 24px 0 12px;line-height:32px;font-size:14px;font-weight:700;color:var(--vp-c-text-1)}.item.appearance[data-v-5ac0399f],.item.social-links[data-v-5ac0399f]{display:flex;align-items:center;padding:0 12px}.item.appearance[data-v-5ac0399f]{min-width:176px}.appearance-action[data-v-5ac0399f]{margin-right:-2px}.social-links-list[data-v-5ac0399f]{margin:-4px -8px}.VPNavBarHamburger[data-v-07c69bee]{display:flex;justify-content:center;align-items:center;width:48px;height:var(--vp-nav-height)}@media (min-width: 768px){.VPNavBarHamburger[data-v-07c69bee]{display:none}}.container[data-v-07c69bee]{position:relative;width:16px;height:14px;overflow:hidden}.VPNavBarHamburger:hover .top[data-v-07c69bee]{top:0;left:0;transform:translate(4px)}.VPNavBarHamburger:hover .middle[data-v-07c69bee]{top:6px;left:0;transform:translate(0)}.VPNavBarHamburger:hover .bottom[data-v-07c69bee]{top:12px;left:0;transform:translate(8px)}.VPNavBarHamburger.active .top[data-v-07c69bee]{top:6px;transform:translate(0) rotate(225deg)}.VPNavBarHamburger.active .middle[data-v-07c69bee]{top:6px;transform:translate(16px)}.VPNavBarHamburger.active .bottom[data-v-07c69bee]{top:6px;transform:translate(0) rotate(135deg)}.VPNavBarHamburger.active:hover .top[data-v-07c69bee],.VPNavBarHamburger.active:hover .middle[data-v-07c69bee],.VPNavBarHamburger.active:hover .bottom[data-v-07c69bee]{background-color:var(--vp-c-text-2);transition:top .25s,background-color .25s,transform .25s}.top[data-v-07c69bee],.middle[data-v-07c69bee],.bottom[data-v-07c69bee]{position:absolute;width:16px;height:2px;background-color:var(--vp-c-text-1);transition:top .25s,background-color .5s,transform .25s}.top[data-v-07c69bee]{top:0;left:0;transform:translate(0)}.middle[data-v-07c69bee]{top:6px;left:0;transform:translate(8px)}.bottom[data-v-07c69bee]{top:12px;left:0;transform:translate(4px)}.VPNavBar[data-v-528d5544]{position:relative;border-bottom:1px solid transparent;padding:0 8px 0 24px;height:var(--vp-nav-height);transition:border-color .5s,background-color .5s;pointer-events:none;white-space:nowrap}.VPNavBar.has-sidebar[data-v-528d5544]{border-bottom-color:var(--vp-c-gutter)}@media (min-width: 768px){.VPNavBar[data-v-528d5544]{padding:0 32px}}@media (min-width: 960px){.VPNavBar.has-sidebar[data-v-528d5544]{border-bottom-color:transparent;padding:0}.VPNavBar.fill[data-v-528d5544]:not(.has-sidebar){border-bottom-color:var(--vp-c-gutter);background-color:var(--vp-nav-bg-color)}}.container[data-v-528d5544]{display:flex;justify-content:space-between;margin:0 auto;max-width:calc(var(--vp-layout-max-width) - 64px);height:var(--vp-nav-height);pointer-events:none}.container>.title[data-v-528d5544],.container>.content[data-v-528d5544]{pointer-events:none}.container[data-v-528d5544] *{pointer-events:auto}@media (min-width: 960px){.VPNavBar.has-sidebar .container[data-v-528d5544]{max-width:100%}}.title[data-v-528d5544]{flex-shrink:0;height:calc(var(--vp-nav-height) - 1px);transition:background-color .5s}@media (min-width: 960px){.VPNavBar.has-sidebar .title[data-v-528d5544]{position:absolute;top:0;left:0;z-index:2;padding:0 32px;width:var(--vp-sidebar-width);height:var(--vp-nav-height);background-color:transparent}}@media (min-width: 1440px){.VPNavBar.has-sidebar .title[data-v-528d5544]{padding-left:max(32px,calc((100% - (var(--vp-layout-max-width) - 64px)) / 2));width:calc((100% - (var(--vp-layout-max-width) - 64px)) / 2 + var(--vp-sidebar-width) - 32px)}}.content[data-v-528d5544]{flex-grow:1}@media (min-width: 960px){.VPNavBar.has-sidebar .content[data-v-528d5544]{position:relative;z-index:1;padding-right:32px;padding-left:var(--vp-sidebar-width)}}@media (min-width: 1440px){.VPNavBar.has-sidebar .content[data-v-528d5544]{padding-right:calc((100vw - var(--vp-layout-max-width)) / 2 + 32px);padding-left:calc((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width))}}.content-body[data-v-528d5544]{display:flex;justify-content:flex-end;align-items:center;height:calc(var(--vp-nav-height) - 1px);transition:background-color .5s}@media (min-width: 960px){.VPNavBar.has-sidebar .content-body[data-v-528d5544],.VPNavBar.fill .content-body[data-v-528d5544]{position:relative;background-color:var(--vp-nav-bg-color)}}.menu+.translations[data-v-528d5544]:before,.menu+.appearance[data-v-528d5544]:before,.menu+.social-links[data-v-528d5544]:before,.translations+.appearance[data-v-528d5544]:before,.appearance+.social-links[data-v-528d5544]:before{margin-right:8px;margin-left:8px;width:1px;height:24px;background-color:var(--vp-c-divider);content:""}.menu+.appearance[data-v-528d5544]:before,.translations+.appearance[data-v-528d5544]:before{margin-right:16px}.appearance+.social-links[data-v-528d5544]:before{margin-left:16px}.social-links[data-v-528d5544]{margin-right:-8px}@media (min-width: 960px){.VPNavBar.has-sidebar .curtain[data-v-528d5544]{position:absolute;right:0;bottom:-31px;width:calc(100% - var(--vp-sidebar-width));height:32px}.VPNavBar.has-sidebar .curtain[data-v-528d5544]:before{display:block;width:100%;height:32px;background:linear-gradient(var(--vp-c-bg),transparent 70%);content:""}}@media (min-width: 1440px){.VPNavBar.has-sidebar .curtain[data-v-528d5544]{width:calc(100% - ((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width)))}}.VPNavScreenMenuLink[data-v-4a38c9c0]{display:block;border-bottom:1px solid var(--vp-c-divider);padding:12px 0 11px;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:border-color .25s,color .25s}.VPNavScreenMenuLink[data-v-4a38c9c0]:hover{color:var(--vp-c-brand)}.VPNavScreenMenuGroupLink[data-v-5b16e6cf]{display:block;margin-left:12px;line-height:32px;font-size:14px;font-weight:400;color:var(--vp-c-text-1);transition:color .25s}.VPNavScreenMenuGroupLink[data-v-5b16e6cf]:hover{color:var(--vp-c-brand)}.VPNavScreenMenuGroupSection[data-v-34625a7e]{display:block}.title[data-v-34625a7e]{line-height:32px;font-size:13px;font-weight:700;color:var(--vp-c-text-2);transition:color .25s}.VPNavScreenMenuGroup[data-v-a3dd94ad]{border-bottom:1px solid var(--vp-c-divider);height:48px;overflow:hidden;transition:border-color .5s}.VPNavScreenMenuGroup .items[data-v-a3dd94ad]{visibility:hidden}.VPNavScreenMenuGroup.open .items[data-v-a3dd94ad]{visibility:visible}.VPNavScreenMenuGroup.open[data-v-a3dd94ad]{padding-bottom:10px;height:auto}.VPNavScreenMenuGroup.open .button[data-v-a3dd94ad]{padding-bottom:6px;color:var(--vp-c-brand)}.VPNavScreenMenuGroup.open .button-icon[data-v-a3dd94ad]{transform:rotate(45deg)}.button[data-v-a3dd94ad]{display:flex;justify-content:space-between;align-items:center;padding:12px 4px 11px 0;width:100%;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.button[data-v-a3dd94ad]:hover{color:var(--vp-c-brand)}.button-icon[data-v-a3dd94ad]{width:14px;height:14px;fill:var(--vp-c-text-2);transition:fill .5s,transform .25s}.group[data-v-a3dd94ad]:first-child{padding-top:0}.group+.group[data-v-a3dd94ad],.group+.item[data-v-a3dd94ad]{padding-top:4px}.VPNavScreenAppearance[data-v-cefb5854]{display:flex;justify-content:space-between;align-items:center;border-radius:8px;padding:12px 14px 12px 16px;background-color:var(--vp-c-bg-soft)}.text[data-v-cefb5854]{line-height:24px;font-size:12px;font-weight:500;color:var(--vp-c-text-2)}.VPNavScreenTranslations[data-v-4d5840eb]{height:24px;overflow:hidden}.VPNavScreenTranslations.open[data-v-4d5840eb]{height:auto}.title[data-v-4d5840eb]{display:flex;align-items:center;font-size:14px;font-weight:500;color:var(--vp-c-text-1)}.icon[data-v-4d5840eb]{width:16px;height:16px;fill:currentColor}.icon.lang[data-v-4d5840eb]{margin-right:8px}.icon.chevron[data-v-4d5840eb]{margin-left:4px}.list[data-v-4d5840eb]{padding:4px 0 0 24px}.link[data-v-4d5840eb]{line-height:32px;font-size:13px;color:var(--vp-c-text-1)}.VPNavScreen[data-v-8337e0f7]{position:fixed;top:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 1px);right:0;bottom:0;left:0;padding:0 32px;width:100%;background-color:var(--vp-nav-screen-bg-color);overflow-y:auto;transition:background-color .5s;pointer-events:auto}.VPNavScreen.fade-enter-active[data-v-8337e0f7],.VPNavScreen.fade-leave-active[data-v-8337e0f7]{transition:opacity .25s}.VPNavScreen.fade-enter-active .container[data-v-8337e0f7],.VPNavScreen.fade-leave-active .container[data-v-8337e0f7]{transition:transform .25s ease}.VPNavScreen.fade-enter-from[data-v-8337e0f7],.VPNavScreen.fade-leave-to[data-v-8337e0f7]{opacity:0}.VPNavScreen.fade-enter-from .container[data-v-8337e0f7],.VPNavScreen.fade-leave-to .container[data-v-8337e0f7]{transform:translateY(-8px)}@media (min-width: 768px){.VPNavScreen[data-v-8337e0f7]{display:none}}.container[data-v-8337e0f7]{margin:0 auto;padding:24px 0 96px;max-width:288px}.menu+.translations[data-v-8337e0f7],.menu+.appearance[data-v-8337e0f7],.translations+.appearance[data-v-8337e0f7]{margin-top:24px}.menu+.social-links[data-v-8337e0f7]{margin-top:16px}.appearance+.social-links[data-v-8337e0f7]{margin-top:16px}.VPNav[data-v-d58b459a]{position:relative;top:var(--vp-layout-top-height, 0px);left:0;z-index:var(--vp-z-index-nav);width:100%;pointer-events:none;transition:background-color .5s}@media (min-width: 960px){.VPNav[data-v-d58b459a]{position:fixed}}.root[data-v-65e3602e]{position:relative;z-index:1}.nested[data-v-65e3602e]{padding-left:13px}.outline-link[data-v-65e3602e]{display:block;line-height:28px;color:var(--vp-c-text-2);white-space:nowrap;overflow:hidden;text-overflow:ellipsis;transition:color .5s;font-weight:500}.outline-link[data-v-65e3602e]:hover,.outline-link.active[data-v-65e3602e]{color:var(--vp-c-text-1);transition:color .25s}.outline-link.nested[data-v-65e3602e]{padding-left:13px}.VPLocalNavOutlineDropdown[data-v-ba7b08b9]{padding:12px 20px 11px}.VPLocalNavOutlineDropdown button[data-v-ba7b08b9]{display:block;font-size:12px;font-weight:500;line-height:24px;color:var(--vp-c-text-2);transition:color .5s;position:relative}.VPLocalNavOutlineDropdown button[data-v-ba7b08b9]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPLocalNavOutlineDropdown button.open[data-v-ba7b08b9]{color:var(--vp-c-text-1)}.icon[data-v-ba7b08b9]{display:inline-block;vertical-align:middle;margin-left:2px;width:14px;height:14px;fill:currentColor}[data-v-ba7b08b9] .outline-link{font-size:14px;padding:2px 0}.open>.icon[data-v-ba7b08b9]{transform:rotate(90deg)}.items[data-v-ba7b08b9]{position:absolute;left:20px;right:20px;top:64px;background-color:var(--vp-local-nav-bg-color);padding:4px 10px 16px;border:1px solid var(--vp-c-divider);border-radius:8px;max-height:calc(var(--vp-vh, 100vh) - 86px);overflow:hidden auto;box-shadow:var(--vp-shadow-3)}.top-link[data-v-ba7b08b9]{display:block;color:var(--vp-c-brand);font-size:13px;font-weight:500;padding:6px 0;margin:0 13px 10px;border-bottom:1px solid var(--vp-c-divider)}.flyout-enter-active[data-v-ba7b08b9]{transition:all .2s ease-out}.flyout-leave-active[data-v-ba7b08b9]{transition:all .15s ease-in}.flyout-enter-from[data-v-ba7b08b9],.flyout-leave-to[data-v-ba7b08b9]{opacity:0;transform:translateY(-16px)}.VPLocalNav[data-v-8c9577da]{position:sticky;top:0;left:0;z-index:var(--vp-z-index-local-nav);display:flex;justify-content:space-between;align-items:center;border-bottom:1px solid var(--vp-c-gutter);padding-top:var(--vp-layout-top-height, 0px);width:100%;background-color:var(--vp-local-nav-bg-color);transition:border-color .5s,background-color .5s}@media (min-width: 960px){.VPLocalNav[data-v-8c9577da]{display:none}}.menu[data-v-8c9577da]{display:flex;align-items:center;padding:12px 24px 11px;line-height:24px;font-size:12px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.menu[data-v-8c9577da]:hover{color:var(--vp-c-text-1);transition:color .25s}@media (min-width: 768px){.menu[data-v-8c9577da]{padding:0 32px}}.menu-icon[data-v-8c9577da]{margin-right:8px;width:16px;height:16px;fill:currentColor}.VPOutlineDropdown[data-v-8c9577da]{padding:12px 24px 11px}@media (min-width: 768px){.VPOutlineDropdown[data-v-8c9577da]{padding:12px 32px 11px}}.VPSidebarItem.level-0[data-v-a2d5c4ca]{padding-bottom:24px}.VPSidebarItem.collapsed.level-0[data-v-a2d5c4ca]{padding-bottom:10px}.item[data-v-a2d5c4ca]{position:relative;display:flex;width:100%}.VPSidebarItem.collapsible>.item[data-v-a2d5c4ca]{cursor:pointer}.indicator[data-v-a2d5c4ca]{position:absolute;top:6px;bottom:6px;left:-17px;width:1px;transition:background-color .25s}.VPSidebarItem.level-2.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-active>.item>.indicator[data-v-a2d5c4ca]{background-color:var(--vp-c-brand)}.link[data-v-a2d5c4ca]{display:flex;align-items:center;flex-grow:1}.text[data-v-a2d5c4ca]{flex-grow:1;padding:4px 0;line-height:24px;font-size:14px;transition:color .25s}.VPSidebarItem.level-0 .text[data-v-a2d5c4ca]{font-weight:700;color:var(--vp-c-text-1)}.VPSidebarItem.level-1 .text[data-v-a2d5c4ca],.VPSidebarItem.level-2 .text[data-v-a2d5c4ca],.VPSidebarItem.level-3 .text[data-v-a2d5c4ca],.VPSidebarItem.level-4 .text[data-v-a2d5c4ca],.VPSidebarItem.level-5 .text[data-v-a2d5c4ca]{font-weight:500;color:var(--vp-c-text-2)}.VPSidebarItem.level-0.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-1.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-2.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-link>.item>.link:hover .text[data-v-a2d5c4ca]{color:var(--vp-c-brand)}.VPSidebarItem.level-0.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-1.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-2.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-3.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-4.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-5.has-active>.item>.link>.text[data-v-a2d5c4ca]{color:var(--vp-c-text-1)}.VPSidebarItem.level-0.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-1.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-2.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-active>.item .link>.text[data-v-a2d5c4ca]{color:var(--vp-c-brand)}.caret[data-v-a2d5c4ca]{display:flex;justify-content:center;align-items:center;margin-right:-7px;width:32px;height:32px;color:var(--vp-c-text-3);cursor:pointer;transition:color .25s}.item:hover .caret[data-v-a2d5c4ca]{color:var(--vp-c-text-2)}.item:hover .caret[data-v-a2d5c4ca]:hover{color:var(--vp-c-text-1)}.caret-icon[data-v-a2d5c4ca]{width:18px;height:18px;fill:currentColor;transform:rotate(90deg);transition:transform .25s}.VPSidebarItem.collapsed .caret-icon[data-v-a2d5c4ca]{transform:rotate(0)}.VPSidebarItem.level-1 .items[data-v-a2d5c4ca],.VPSidebarItem.level-2 .items[data-v-a2d5c4ca],.VPSidebarItem.level-3 .items[data-v-a2d5c4ca],.VPSidebarItem.level-4 .items[data-v-a2d5c4ca],.VPSidebarItem.level-5 .items[data-v-a2d5c4ca]{border-left:1px solid var(--vp-c-divider);padding-left:16px}.VPSidebarItem.collapsed .items[data-v-a2d5c4ca]{display:none}.VPSidebar[data-v-c7a4f3fe]{position:fixed;top:var(--vp-layout-top-height, 0px);bottom:0;left:0;z-index:var(--vp-z-index-sidebar);padding:32px 32px 96px;width:calc(100vw - 64px);max-width:320px;background-color:var(--vp-sidebar-bg-color);opacity:0;box-shadow:var(--vp-c-shadow-3);overflow-x:hidden;overflow-y:auto;transform:translate(-100%);transition:opacity .5s,transform .25s ease;overscroll-behavior:contain}.VPSidebar.open[data-v-c7a4f3fe]{opacity:1;visibility:visible;transform:translate(0);transition:opacity .25s,transform .5s cubic-bezier(.19,1,.22,1)}.dark .VPSidebar[data-v-c7a4f3fe]{box-shadow:var(--vp-shadow-1)}@media (min-width: 960px){.VPSidebar[data-v-c7a4f3fe]{z-index:1;padding-top:var(--vp-nav-height);padding-bottom:128px;width:var(--vp-sidebar-width);max-width:100%;background-color:var(--vp-sidebar-bg-color);opacity:1;visibility:visible;box-shadow:none;transform:translate(0)}}@media (min-width: 1440px){.VPSidebar[data-v-c7a4f3fe]{padding-left:max(32px,calc((100% - (var(--vp-layout-max-width) - 64px)) / 2));width:calc((100% - (var(--vp-layout-max-width) - 64px)) / 2 + var(--vp-sidebar-width) - 32px)}}@media (min-width: 960px){.curtain[data-v-c7a4f3fe]{position:sticky;top:-64px;left:0;z-index:1;margin-top:calc(var(--vp-nav-height) * -1);margin-right:-32px;margin-left:-32px;height:var(--vp-nav-height);background-color:var(--vp-sidebar-bg-color)}}.nav[data-v-c7a4f3fe]{outline:0}.group+.group[data-v-c7a4f3fe]{border-top:1px solid var(--vp-c-divider);padding-top:10px}@media (min-width: 960px){.group[data-v-c7a4f3fe]{padding-top:10px;width:calc(var(--vp-sidebar-width) - 64px)}}.VPButton[data-v-30800ca8]{display:inline-block;border:1px solid transparent;text-align:center;font-weight:600;white-space:nowrap;transition:color .25s,border-color .25s,background-color .25s}.VPButton[data-v-30800ca8]:active{transition:color .1s,border-color .1s,background-color .1s}.VPButton.medium[data-v-30800ca8]{border-radius:20px;padding:0 20px;line-height:38px;font-size:14px}.VPButton.big[data-v-30800ca8]{border-radius:24px;padding:0 24px;line-height:46px;font-size:16px}.VPButton.brand[data-v-30800ca8]{border-color:var(--vp-button-brand-border);color:var(--vp-button-brand-text);background-color:var(--vp-button-brand-bg)}.VPButton.brand[data-v-30800ca8]:hover{border-color:var(--vp-button-brand-hover-border);color:var(--vp-button-brand-hover-text);background-color:var(--vp-button-brand-hover-bg)}.VPButton.brand[data-v-30800ca8]:active{border-color:var(--vp-button-brand-active-border);color:var(--vp-button-brand-active-text);background-color:var(--vp-button-brand-active-bg)}.VPButton.alt[data-v-30800ca8]{border-color:var(--vp-button-alt-border);color:var(--vp-button-alt-text);background-color:var(--vp-button-alt-bg)}.VPButton.alt[data-v-30800ca8]:hover{border-color:var(--vp-button-alt-hover-border);color:var(--vp-button-alt-hover-text);background-color:var(--vp-button-alt-hover-bg)}.VPButton.alt[data-v-30800ca8]:active{border-color:var(--vp-button-alt-active-border);color:var(--vp-button-alt-active-text);background-color:var(--vp-button-alt-active-bg)}.VPButton.sponsor[data-v-30800ca8]{border-color:var(--vp-button-sponsor-border);color:var(--vp-button-sponsor-text);background-color:var(--vp-button-sponsor-bg)}.VPButton.sponsor[data-v-30800ca8]:hover{border-color:var(--vp-button-sponsor-hover-border);color:var(--vp-button-sponsor-hover-text);background-color:var(--vp-button-sponsor-hover-bg)}.VPButton.sponsor[data-v-30800ca8]:active{border-color:var(--vp-button-sponsor-active-border);color:var(--vp-button-sponsor-active-text);background-color:var(--vp-button-sponsor-active-bg)}.VPHero[data-v-9e7a9339]{margin-top:calc((var(--vp-nav-height) + var(--vp-layout-top-height, 0px)) * -1);padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 48px) 24px 48px}@media (min-width: 640px){.VPHero[data-v-9e7a9339]{padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 80px) 48px 64px}}@media (min-width: 960px){.VPHero[data-v-9e7a9339]{padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 80px) 64px 64px}}.container[data-v-9e7a9339]{display:flex;flex-direction:column;margin:0 auto;max-width:1152px}@media (min-width: 960px){.container[data-v-9e7a9339]{flex-direction:row}}.main[data-v-9e7a9339]{position:relative;z-index:10;order:2;flex-grow:1;flex-shrink:0}.VPHero.has-image .container[data-v-9e7a9339]{text-align:center}@media (min-width: 960px){.VPHero.has-image .container[data-v-9e7a9339]{text-align:left}}@media (min-width: 960px){.main[data-v-9e7a9339]{order:1;width:calc((100% / 3) * 2)}.VPHero.has-image .main[data-v-9e7a9339]{max-width:592px}}.name[data-v-9e7a9339],.text[data-v-9e7a9339]{max-width:392px;letter-spacing:-.4px;line-height:40px;font-size:32px;font-weight:700;white-space:pre-wrap}.VPHero.has-image .name[data-v-9e7a9339],.VPHero.has-image .text[data-v-9e7a9339]{margin:0 auto}.name[data-v-9e7a9339]{color:var(--vp-home-hero-name-color)}.clip[data-v-9e7a9339]{background:var(--vp-home-hero-name-background);-webkit-background-clip:text;background-clip:text;-webkit-text-fill-color:var(--vp-home-hero-name-color)}@media (min-width: 640px){.name[data-v-9e7a9339],.text[data-v-9e7a9339]{max-width:576px;line-height:56px;font-size:48px}}@media (min-width: 960px){.name[data-v-9e7a9339],.text[data-v-9e7a9339]{line-height:64px;font-size:56px}.VPHero.has-image .name[data-v-9e7a9339],.VPHero.has-image .text[data-v-9e7a9339]{margin:0}}.tagline[data-v-9e7a9339]{padding-top:8px;max-width:392px;line-height:28px;font-size:18px;font-weight:500;white-space:pre-wrap;color:var(--vp-c-text-2)}.VPHero.has-image .tagline[data-v-9e7a9339]{margin:0 auto}@media (min-width: 640px){.tagline[data-v-9e7a9339]{padding-top:12px;max-width:576px;line-height:32px;font-size:20px}}@media (min-width: 960px){.tagline[data-v-9e7a9339]{line-height:36px;font-size:24px}.VPHero.has-image .tagline[data-v-9e7a9339]{margin:0}}.actions[data-v-9e7a9339]{display:flex;flex-wrap:wrap;margin:-6px;padding-top:24px}.VPHero.has-image .actions[data-v-9e7a9339]{justify-content:center}@media (min-width: 640px){.actions[data-v-9e7a9339]{padding-top:32px}}@media (min-width: 960px){.VPHero.has-image .actions[data-v-9e7a9339]{justify-content:flex-start}}.action[data-v-9e7a9339]{flex-shrink:0;padding:6px}.image[data-v-9e7a9339]{order:1;margin:-76px -24px -48px}@media (min-width: 640px){.image[data-v-9e7a9339]{margin:-108px -24px -48px}}@media (min-width: 960px){.image[data-v-9e7a9339]{flex-grow:1;order:2;margin:0;min-height:100%}}.image-container[data-v-9e7a9339]{position:relative;margin:0 auto;width:320px;height:320px}@media (min-width: 640px){.image-container[data-v-9e7a9339]{width:392px;height:392px}}@media (min-width: 960px){.image-container[data-v-9e7a9339]{display:flex;justify-content:center;align-items:center;width:100%;height:100%;transform:translate(-32px,-32px)}}.image-bg[data-v-9e7a9339]{position:absolute;top:50%;left:50%;border-radius:50%;width:192px;height:192px;background-image:var(--vp-home-hero-image-background-image);filter:var(--vp-home-hero-image-filter);transform:translate(-50%,-50%)}@media (min-width: 640px){.image-bg[data-v-9e7a9339]{width:256px;height:256px}}@media (min-width: 960px){.image-bg[data-v-9e7a9339]{width:320px;height:320px}}[data-v-9e7a9339] .image-src{position:absolute;top:50%;left:50%;max-width:192px;max-height:192px;transform:translate(-50%,-50%)}@media (min-width: 640px){[data-v-9e7a9339] .image-src{max-width:256px;max-height:256px}}@media (min-width: 960px){[data-v-9e7a9339] .image-src{max-width:320px;max-height:320px}}.VPFeature[data-v-c4ab7714]{display:block;border:1px solid var(--vp-c-bg-soft);border-radius:12px;height:100%;background-color:var(--vp-c-bg-soft);transition:border-color .25s,background-color .25s}.VPFeature.link[data-v-c4ab7714]:hover{border-color:var(--vp-c-brand);background-color:var(--vp-c-bg-soft-up)}.box[data-v-c4ab7714]{display:flex;flex-direction:column;padding:24px;height:100%}.VPFeature[data-v-c4ab7714] .VPImage{width:48px;height:48px;margin-bottom:20px}.icon[data-v-c4ab7714]{display:flex;justify-content:center;align-items:center;margin-bottom:20px;border-radius:6px;background-color:var(--vp-c-bg-soft-down);width:48px;height:48px;font-size:24px;transition:background-color .25s}.title[data-v-c4ab7714]{line-height:24px;font-size:16px;font-weight:600}.details[data-v-c4ab7714]{flex-grow:1;padding-top:8px;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.link-text[data-v-c4ab7714]{padding-top:8px}.link-text-value[data-v-c4ab7714]{display:flex;align-items:center;font-size:14px;font-weight:500;color:var(--vp-c-brand)}.link-text-icon[data-v-c4ab7714]{display:inline-block;margin-left:6px;width:14px;height:14px;fill:currentColor}.VPFeatures[data-v-1cc0eb36]{position:relative;padding:0 24px}@media (min-width: 640px){.VPFeatures[data-v-1cc0eb36]{padding:0 48px}}@media (min-width: 960px){.VPFeatures[data-v-1cc0eb36]{padding:0 64px}}.container[data-v-1cc0eb36]{margin:0 auto;max-width:1152px}.items[data-v-1cc0eb36]{display:flex;flex-wrap:wrap;margin:-8px}.item[data-v-1cc0eb36]{padding:8px;width:100%}@media (min-width: 640px){.item.grid-2[data-v-1cc0eb36],.item.grid-4[data-v-1cc0eb36],.item.grid-6[data-v-1cc0eb36]{width:50%}}@media (min-width: 768px){.item.grid-2[data-v-1cc0eb36],.item.grid-4[data-v-1cc0eb36]{width:50%}.item.grid-3[data-v-1cc0eb36],.item.grid-6[data-v-1cc0eb36]{width:calc(100% / 3)}}@media (min-width: 960px){.item.grid-4[data-v-1cc0eb36]{width:25%}}.VPHome[data-v-0cc58ac0]{padding-bottom:96px}.VPHome[data-v-0cc58ac0] .VPHomeSponsors{margin-top:112px;margin-bottom:-128px}@media (min-width: 768px){.VPHome[data-v-0cc58ac0]{padding-bottom:128px}}.VPDocAsideOutline[data-v-7c84bbba]{display:none}.VPDocAsideOutline.has-outline[data-v-7c84bbba]{display:block}.content[data-v-7c84bbba]{position:relative;border-left:1px solid var(--vp-c-divider);padding-left:16px;font-size:13px;font-weight:500}.outline-marker[data-v-7c84bbba]{position:absolute;top:32px;left:-1px;z-index:0;opacity:0;width:1px;height:18px;background-color:var(--vp-c-brand);transition:top .25s cubic-bezier(0,1,.5,1),background-color .5s,opacity .25s}.outline-title[data-v-7c84bbba]{letter-spacing:.4px;line-height:28px;font-size:13px;font-weight:600}.VPDocAside[data-v-ac259e18]{display:flex;flex-direction:column;flex-grow:1}.spacer[data-v-ac259e18]{flex-grow:1}.VPDocAside[data-v-ac259e18] .spacer+.VPDocAsideSponsors,.VPDocAside[data-v-ac259e18] .spacer+.VPDocAsideCarbonAds{margin-top:24px}.VPDocAside[data-v-ac259e18] .VPDocAsideSponsors+.VPDocAsideCarbonAds{margin-top:16px}.VPLastUpdated[data-v-d35b8cb4]{line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}@media (min-width: 640px){.VPLastUpdated[data-v-d35b8cb4]{line-height:32px;font-size:14px;font-weight:500}}.VPDocFooter[data-v-2b4f87e4]{margin-top:64px}.edit-info[data-v-2b4f87e4]{padding-bottom:18px}@media (min-width: 640px){.edit-info[data-v-2b4f87e4]{display:flex;justify-content:space-between;align-items:center;padding-bottom:14px}}.edit-link-button[data-v-2b4f87e4]{display:flex;align-items:center;border:0;line-height:32px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:color .25s}.edit-link-button[data-v-2b4f87e4]:hover{color:var(--vp-c-brand-dark)}.edit-link-icon[data-v-2b4f87e4]{margin-right:8px;width:14px;height:14px;fill:currentColor}.prev-next[data-v-2b4f87e4]{border-top:1px solid var(--vp-c-divider);padding-top:24px}@media (min-width: 640px){.prev-next[data-v-2b4f87e4]{display:flex}}.pager.has-prev[data-v-2b4f87e4]{padding-top:8px}@media (min-width: 640px){.pager[data-v-2b4f87e4]{display:flex;flex-direction:column;flex-shrink:0;width:50%}.pager.has-prev[data-v-2b4f87e4]{padding-top:0;padding-left:16px}}.pager-link[data-v-2b4f87e4]{display:block;border:1px solid var(--vp-c-divider);border-radius:8px;padding:11px 16px 13px;width:100%;height:100%;transition:border-color .25s}.pager-link[data-v-2b4f87e4]:hover{border-color:var(--vp-c-brand)}.pager-link.next[data-v-2b4f87e4]{margin-left:auto;text-align:right}.desc[data-v-2b4f87e4]{display:block;line-height:20px;font-size:12px;font-weight:500;color:var(--vp-c-text-2)}.title[data-v-2b4f87e4]{display:block;line-height:20px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:color .25s}.VPDocOutlineDropdown[data-v-7fff1eb3]{margin-bottom:42px}.VPDocOutlineDropdown button[data-v-7fff1eb3]{display:block;font-size:14px;font-weight:500;line-height:24px;color:var(--vp-c-text-2);transition:color .5s;border:1px solid var(--vp-c-border);padding:4px 12px;border-radius:8px}.VPDocOutlineDropdown button[data-v-7fff1eb3]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPDocOutlineDropdown button.open[data-v-7fff1eb3]{color:var(--vp-c-text-1)}.icon[data-v-7fff1eb3]{display:inline-block;vertical-align:middle;margin-left:2px;width:14px;height:14px;fill:currentColor}[data-v-7fff1eb3] .outline-link{font-size:13px}.open>.icon[data-v-7fff1eb3]{transform:rotate(90deg)}.items[data-v-7fff1eb3]{margin-top:10px;border-left:1px solid var(--vp-c-divider)}.VPDoc[data-v-4c07d6a3]{padding:32px 24px 96px;width:100%}.VPDoc .VPDocOutlineDropdown[data-v-4c07d6a3]{display:none}@media (min-width: 960px) and (max-width: 1280px){.VPDoc .VPDocOutlineDropdown[data-v-4c07d6a3]{display:block}}@media (min-width: 768px){.VPDoc[data-v-4c07d6a3]{padding:48px 32px 128px}}@media (min-width: 960px){.VPDoc[data-v-4c07d6a3]{padding:32px 32px 0}.VPDoc:not(.has-sidebar) .container[data-v-4c07d6a3]{display:flex;justify-content:center;max-width:992px}.VPDoc:not(.has-sidebar) .content[data-v-4c07d6a3]{max-width:752px}}@media (min-width: 1280px){.VPDoc .container[data-v-4c07d6a3]{display:flex;justify-content:center}.VPDoc .aside[data-v-4c07d6a3]{display:block}}@media (min-width: 1440px){.VPDoc:not(.has-sidebar) .content[data-v-4c07d6a3]{max-width:784px}.VPDoc:not(.has-sidebar) .container[data-v-4c07d6a3]{max-width:1104px}}.container[data-v-4c07d6a3]{margin:0 auto;width:100%}.aside[data-v-4c07d6a3]{position:relative;display:none;order:2;flex-grow:1;padding-left:32px;width:100%;max-width:256px}.left-aside[data-v-4c07d6a3]{order:1;padding-left:unset;padding-right:32px}.aside-container[data-v-4c07d6a3]{position:fixed;top:0;padding-top:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + var(--vp-doc-top-height, 0px) + 32px);width:224px;height:100vh;overflow-x:hidden;overflow-y:auto;scrollbar-width:none}.aside-container[data-v-4c07d6a3]::-webkit-scrollbar{display:none}.aside-curtain[data-v-4c07d6a3]{position:fixed;bottom:0;z-index:10;width:224px;height:32px;background:linear-gradient(transparent,var(--vp-c-bg) 70%)}.aside-content[data-v-4c07d6a3]{display:flex;flex-direction:column;min-height:calc(100vh - (var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 32px));padding-bottom:32px}.content[data-v-4c07d6a3]{position:relative;margin:0 auto;width:100%}@media (min-width: 960px){.content[data-v-4c07d6a3]{padding:0 32px 128px}}@media (min-width: 1280px){.content[data-v-4c07d6a3]{order:1;margin:0;min-width:640px}}.content-container[data-v-4c07d6a3]{margin:0 auto}.VPDoc.has-aside .content-container[data-v-4c07d6a3]{max-width:688px}.NotFound[data-v-135d8cce]{padding:64px 24px 96px;text-align:center}@media (min-width: 768px){.NotFound[data-v-135d8cce]{padding:96px 32px 168px}}.code[data-v-135d8cce]{line-height:64px;font-size:64px;font-weight:600}.title[data-v-135d8cce]{padding-top:12px;letter-spacing:2px;line-height:20px;font-size:20px;font-weight:700}.divider[data-v-135d8cce]{margin:24px auto 18px;width:64px;height:1px;background-color:var(--vp-c-divider)}.quote[data-v-135d8cce]{margin:0 auto;max-width:256px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.action[data-v-135d8cce]{padding-top:20px}.link[data-v-135d8cce]{display:inline-block;border:1px solid var(--vp-c-brand);border-radius:16px;padding:3px 16px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:border-color .25s,color .25s}.link[data-v-135d8cce]:hover{border-color:var(--vp-c-brand-dark);color:var(--vp-c-brand-dark)}.VPContent[data-v-185f5406]{flex-grow:1;flex-shrink:0;margin:var(--vp-layout-top-height, 0px) auto 0;width:100%}.VPContent.is-home[data-v-185f5406]{width:100%;max-width:100%}.VPContent.has-sidebar[data-v-185f5406]{margin:0}@media (min-width: 960px){.VPContent[data-v-185f5406]{padding-top:var(--vp-nav-height)}.VPContent.has-sidebar[data-v-185f5406]{margin:var(--vp-layout-top-height, 0px) 0 0;padding-left:var(--vp-sidebar-width)}}@media (min-width: 1440px){.VPContent.has-sidebar[data-v-185f5406]{padding-right:calc((100vw - var(--vp-layout-max-width)) / 2);padding-left:calc((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width))}}.VPFooter[data-v-e35069ff]{position:relative;z-index:var(--vp-z-index-footer);border-top:1px solid var(--vp-c-gutter);padding:32px 24px;background-color:var(--vp-c-bg)}.VPFooter.has-sidebar[data-v-e35069ff]{display:none}@media (min-width: 768px){.VPFooter[data-v-e35069ff]{padding:32px}}.container[data-v-e35069ff]{margin:0 auto;max-width:var(--vp-layout-max-width);text-align:center}.message[data-v-e35069ff],.copyright[data-v-e35069ff]{line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.message[data-v-e35069ff]{order:2}.copyright[data-v-e35069ff]{order:1}.Layout[data-v-b4389d08]{display:flex;flex-direction:column;min-height:100vh}.VPHomeSponsors[data-v-c6a38a06]{border-top:1px solid var(--vp-c-gutter);padding:88px 24px 96px;background-color:var(--vp-c-bg)}.container[data-v-c6a38a06]{margin:0 auto;max-width:1152px}.love[data-v-c6a38a06]{margin:0 auto;width:28px;height:28px;color:var(--vp-c-text-3)}.icon[data-v-c6a38a06]{width:28px;height:28px;fill:currentColor}.message[data-v-c6a38a06]{margin:0 auto;padding-top:10px;max-width:320px;text-align:center;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}.sponsors[data-v-c6a38a06]{padding-top:32px}.action[data-v-c6a38a06]{padding-top:40px;text-align:center}.VPTeamPage[data-v-77c11d23]{padding-bottom:96px}@media (min-width: 768px){.VPTeamPage[data-v-77c11d23]{padding-bottom:128px}}.VPTeamPageSection+.VPTeamPageSection[data-v-77c11d23-s],.VPTeamMembers+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:64px}.VPTeamMembers+.VPTeamMembers[data-v-77c11d23-s]{margin-top:24px}@media (min-width: 768px){.VPTeamPageTitle+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:16px}.VPTeamPageSection+.VPTeamPageSection[data-v-77c11d23-s],.VPTeamMembers+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:96px}}.VPTeamMembers[data-v-77c11d23-s]{padding:0 24px}@media (min-width: 768px){.VPTeamMembers[data-v-77c11d23-s]{padding:0 48px}}@media (min-width: 960px){.VPTeamMembers[data-v-77c11d23-s]{padding:0 64px}}.VPTeamPageTitle[data-v-9bd857d2]{padding:48px 32px;text-align:center}@media (min-width: 768px){.VPTeamPageTitle[data-v-9bd857d2]{padding:64px 48px 48px}}@media (min-width: 960px){.VPTeamPageTitle[data-v-9bd857d2]{padding:80px 64px 48px}}.title[data-v-9bd857d2]{letter-spacing:0;line-height:44px;font-size:36px;font-weight:500}@media (min-width: 768px){.title[data-v-9bd857d2]{letter-spacing:-.5px;line-height:56px;font-size:48px}}.lead[data-v-9bd857d2]{margin:0 auto;max-width:512px;padding-top:12px;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}@media (min-width: 768px){.lead[data-v-9bd857d2]{max-width:592px;letter-spacing:.15px;line-height:28px;font-size:20px}}.VPTeamPageSection[data-v-ed31233b]{padding:0 32px}@media (min-width: 768px){.VPTeamPageSection[data-v-ed31233b]{padding:0 48px}}@media (min-width: 960px){.VPTeamPageSection[data-v-ed31233b]{padding:0 64px}}.title[data-v-ed31233b]{position:relative;margin:0 auto;max-width:1152px;text-align:center;color:var(--vp-c-text-2)}.title-line[data-v-ed31233b]{position:absolute;top:16px;left:0;width:100%;height:1px;background-color:var(--vp-c-divider)}.title-text[data-v-ed31233b]{position:relative;display:inline-block;padding:0 24px;letter-spacing:0;line-height:32px;font-size:20px;font-weight:500;background-color:var(--vp-c-bg)}.lead[data-v-ed31233b]{margin:0 auto;max-width:480px;padding-top:12px;text-align:center;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}.members[data-v-ed31233b]{padding-top:40px}.VPTeamMembersItem[data-v-2c315693]{display:flex;flex-direction:column;gap:2px;border-radius:12px;width:100%;height:100%;overflow:hidden}.VPTeamMembersItem.small .profile[data-v-2c315693]{padding:32px}.VPTeamMembersItem.small .data[data-v-2c315693]{padding-top:20px}.VPTeamMembersItem.small .avatar[data-v-2c315693]{width:64px;height:64px}.VPTeamMembersItem.small .name[data-v-2c315693]{line-height:24px;font-size:16px}.VPTeamMembersItem.small .affiliation[data-v-2c315693]{padding-top:4px;line-height:20px;font-size:14px}.VPTeamMembersItem.small .desc[data-v-2c315693]{padding-top:12px;line-height:20px;font-size:14px}.VPTeamMembersItem.small .links[data-v-2c315693]{margin:0 -16px -20px;padding:10px 0 0}.VPTeamMembersItem.medium .profile[data-v-2c315693]{padding:48px 32px}.VPTeamMembersItem.medium .data[data-v-2c315693]{padding-top:24px;text-align:center}.VPTeamMembersItem.medium .avatar[data-v-2c315693]{width:96px;height:96px}.VPTeamMembersItem.medium .name[data-v-2c315693]{letter-spacing:.15px;line-height:28px;font-size:20px}.VPTeamMembersItem.medium .affiliation[data-v-2c315693]{padding-top:4px;font-size:16px}.VPTeamMembersItem.medium .desc[data-v-2c315693]{padding-top:16px;max-width:288px;font-size:16px}.VPTeamMembersItem.medium .links[data-v-2c315693]{margin:0 -16px -12px;padding:16px 12px 0}.profile[data-v-2c315693]{flex-grow:1;background-color:var(--vp-c-bg-soft)}.data[data-v-2c315693]{text-align:center}.avatar[data-v-2c315693]{position:relative;flex-shrink:0;margin:0 auto;border-radius:50%;box-shadow:var(--vp-shadow-3)}.avatar-img[data-v-2c315693]{position:absolute;top:0;right:0;bottom:0;left:0;border-radius:50%;object-fit:cover}.name[data-v-2c315693]{margin:0;font-weight:600}.affiliation[data-v-2c315693]{margin:0;font-weight:500;color:var(--vp-c-text-2)}.org.link[data-v-2c315693]{color:var(--vp-c-text-2);transition:color .25s}.org.link[data-v-2c315693]:hover{color:var(--vp-c-brand)}.desc[data-v-2c315693]{margin:0 auto}.desc[data-v-2c315693] a{font-weight:500;color:var(--vp-c-brand);text-decoration-style:dotted;transition:color .25s}.links[data-v-2c315693]{display:flex;justify-content:center;height:56px}.sp-link[data-v-2c315693]{display:flex;justify-content:center;align-items:center;text-align:center;padding:16px;font-size:14px;font-weight:500;color:var(--vp-c-sponsor);background-color:var(--vp-c-bg-soft);transition:color .25s,background-color .25s}.sp .sp-link.link[data-v-2c315693]:hover,.sp .sp-link.link[data-v-2c315693]:focus{outline:none;color:var(--vp-c-white);background-color:var(--vp-c-sponsor)}.sp-icon[data-v-2c315693]{margin-right:8px;width:16px;height:16px;fill:currentColor}.VPTeamMembers.small .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(224px,1fr))}.VPTeamMembers.small.count-1 .container[data-v-9fb4c5f4]{max-width:276px}.VPTeamMembers.small.count-2 .container[data-v-9fb4c5f4]{max-width:576px}.VPTeamMembers.small.count-3 .container[data-v-9fb4c5f4]{max-width:876px}.VPTeamMembers.medium .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(256px,1fr))}@media (min-width: 375px){.VPTeamMembers.medium .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(288px,1fr))}}.VPTeamMembers.medium.count-1 .container[data-v-9fb4c5f4]{max-width:368px}.VPTeamMembers.medium.count-2 .container[data-v-9fb4c5f4]{max-width:760px}.container[data-v-9fb4c5f4]{display:grid;gap:24px;margin:0 auto;max-width:1152px}#nprogress{pointer-events:none}#nprogress .bar{background:var(--vp-c-brand);position:fixed;z-index:1031;top:0;left:0;width:100%;height:2px}#nprogress .peg{display:block;position:absolute;right:0px;width:100px;height:100%;box-shadow:0 0 10px var(--vp-c-brand),0 0 5px var(--vp-c-brand);opacity:1;-webkit-transform:rotate(3deg) translate(0px,-4px);-ms-transform:rotate(3deg) translate(0px,-4px);transform:rotate(3deg) translateY(-4px)}#nprogress .spinner{display:block;position:fixed;z-index:1031;top:15px;right:15px}#nprogress .spinner-icon{width:18px;height:18px;box-sizing:border-box;border:solid 2px transparent;border-top-color:var(--vp-c-brand);border-left-color:var(--vp-c-brand);border-radius:50%;-webkit-animation:nprogress-spinner .4s linear infinite;animation:nprogress-spinner .4s linear infinite}.nprogress-custom-parent{overflow:hidden;position:relative}.nprogress-custom-parent #nprogress .spinner,.nprogress-custom-parent #nprogress .bar{position:absolute}@-webkit-keyframes nprogress-spinner{0%{-webkit-transform:rotate(0deg)}to{-webkit-transform:rotate(360deg)}}@keyframes nprogress-spinner{0%{transform:rotate(0)}to{transform:rotate(360deg)}}*{scrollbar-color:var(--vp-c-divider-light) var(--vp-button-alt-bg);scrollbar-width:thin}::-webkit-scrollbar{width:8px;height:8px}::-webkit-scrollbar-thumb{background:var(--vp-c-mute)}::-webkit-scrollbar-thumb:hover{background:var(--vp-button-alt-hover-bg)}::-webkit-scrollbar-track{background:var(--vp-button-alt-bg)}::-webkit-scrollbar-corner{background:transparent}.katex{font-size:1.05em;direction:ltr}.katex-display{overflow:auto hidden;-webkit-overflow-scrolling:touch;padding-top:.2em;padding-bottom:.2em}.katex-display::-webkit-scrollbar{height:3px}.katex-display .katex{font-size:1.21em}.katex-error{color:red}.medium-zoom-overlay{z-index:20}.medium-zoom-image{z-index:21}.medium-zoom-image{align-items:center}.VPDoc .content-container img{border-radius:5px}.heart{animation:heart 5s ease infinite;color:red}@keyframes heart{0%{transform:scale(1)}5%{transform:scale(1.25)}20%{transform:scale(1)}30%{transform:scale(1)}35%{transform:scale(1.25)}50%{transform:scale(1)}55%{transform:scale(1.25)}70%{transform:scale(1)}}.coffee{color:#580d0d}pre,code,kbd,samp{font-family:var(--vp-font-family-mono)}img.VPImage.image-src{width:300px}.vp-doc>div{width:100%}@media (min-width: 1440px){.VPDoc.has-aside .content-container.content-container{max-width:100%}}.vp-doc p a+a{margin-left:17px}@media (max-width: 1080px){:root{--vp-sidebar-width: 260px}.VPNav .DocSearch-Button-Placeholder{display:none}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:10px;padding-right:10px}}@media (max-width: 992px){.VPNav .VPNavBar{padding-left:24px;padding-right:24px}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:8px;padding-right:8px}}@media (max-width: 800px){.VPNav .VPNavBarSearch{padding-left:16px}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:6px;padding-right:6px}}.VPMenuGroup .title{white-space:nowrap}.DocSearch-Logo{display:none}.link-preview-widget{display:table;width:100%;border:2px solid rgba(55,53,47,.16);border-radius:5px}.link-preview-widget-title{font-size:16px;font-weight:700;display:-webkit-box;overflow:hidden;margin-bottom:8px;word-break:break-all;-webkit-box-orient:vertical;-webkit-line-clamp:2}.link-preview-widget-description{font-size:12px;font-style:normal;line-height:1.5;display:-webkit-box;overflow:hidden;max-height:3em;margin-bottom:4px;word-break:break-all;color:#37352fa6;-webkit-box-orient:vertical;-webkit-line-clamp:2}.link-preview-widget-url{font-size:12px;font-style:normal;line-height:1.5;display:block;margin-bottom:0;color:#37352fa6}.link-preview-widget>a{text-decoration:none!important;display:table-cell;flex-direction:column;padding:16px;cursor:pointer;vertical-align:middle;color:inherit;background-color:transparent;-webkit-box-orient:vertical;-webkit-box-direction:normal;-webkit-box-flex:1;flex:1;transition:background-color .2s ease-in-out;border-radius:5px}.link-preview-widget>a:hover{background-color:var(--vp-custom-block-tip-bg)}.link-preview-widget-image{width:260px;min-width:220px;height:150px;padding:0;vertical-align:middle;border-left:1px rgba(55,53,47,.16) #f2f2f2;border-radius:5px;background-repeat:no-repeat;background-position:50%;background-size:cover;-webkit-box-flex:0;flex:0}html.dark .link-preview-widget{display:table;width:100%;border:2px solid rgba(200,202,208,.16);border-radius:5px}html.dark .link-preview-widget-title{font-size:16px;font-weight:700;display:-webkit-box;overflow:hidden;margin-bottom:8px;word-break:break-all;-webkit-box-orient:vertical;-webkit-line-clamp:2}html.dark .link-preview-widget-description{font-size:12px;font-style:normal;line-height:1.5;display:-webkit-box;overflow:hidden;max-height:3em;margin-bottom:4px;word-break:break-all;color:var(--vp-c-text-2);-webkit-box-orient:vertical;-webkit-line-clamp:2}html.dark .link-preview-widget-url{font-size:12px;font-style:normal;line-height:1.5;display:block;margin-bottom:0;color:var(--vp-c-text-2)}html.dark .link-preview-widget>a{text-decoration:none!important;display:table-cell;flex-direction:column;padding:16px;cursor:pointer;vertical-align:middle;color:inherit;background-color:transparent;-webkit-box-orient:vertical;-webkit-box-direction:normal;-webkit-box-flex:1;flex:1;transition:background-color .2s ease-in-out}html.dark .link-preview-widget>a:hover{background-color:var(--vp-custom-block-tip-bg)}html.dark .link-preview-widget-image{width:260px;min-width:220px;height:150px;padding:0;vertical-align:middle;border-left:1px rgba(200,202,208,.16) #f2f2f2;border-radius:5px;background-repeat:no-repeat;background-position:50%;background-size:cover;-webkit-box-flex:0;flex:0}:root{--vp-home-hero-name-color: transparent;--vp-home-hero-name-background: -webkit-linear-gradient(315deg, #42d392 25%, #647eff);--vp-home-hero-image-background-image: linear-gradient(-45deg, #42d392 50%, #47caff 50%);--vp-home-hero-image-filter: blur(40px)}@media (min-width: 640px){:root{--vp-home-hero-image-filter: blur(56px)}}@media (min-width: 960px){:root{--vp-home-hero-image-filter: blur(72px)}}.vp-sponsor-grid.medium .vp-sponsor-grid-image{max-height:90px}.copyright[data-v-ddda9334]{border-color:var(--vp-custom-block-tip-border);border-style:solid;border-width:1px;background-color:var(--vp-custom-block-tip-bg);border-radius:6px;color:var(--vp-c-text-2);font-size:15px;margin-top:50px}.copyright .content[data-v-ddda9334]{padding:13px 16px}.copyright .content .item[data-v-ddda9334]{margin-bottom:5px;word-break:break-word;line-height:22px}.copyright .content .item .icon[data-v-ddda9334]{display:inline-block;height:16px;width:16px;margin-right:.375rem;vertical-align:-2.5px}a[data-v-ddda9334]{font-weight:400;color:var(--vp-c-text-2)}a[data-v-ddda9334]:hover{color:var(--vp-c-brand)}.copyrightText[data-v-ddda9334]{font-weight:700}/*! normalize.css v8.0.1 | MIT License | github.com/necolas/normalize.css */html{line-height:1.15;-webkit-text-size-adjust:100%;-webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale}body{margin:0}main{display:block}h1{margin:.67em 0;font-size:2em}hr{box-sizing:content-box;height:0;overflow:visible}pre{font-size:1em;font-family:monospace,monospace}a{background-color:transparent}abbr[title]{text-decoration:underline;text-decoration:underline dotted;border-bottom:none}b,strong{font-weight:bolder}code,kbd,samp{font-size:1em;font-family:monospace,monospace}small{font-size:80%}sub,sup{position:relative;font-size:75%;line-height:0;vertical-align:baseline}sub{bottom:-.25em}sup{top:-.5em}img{border-style:none}button,input,optgroup,select,textarea{margin:0;font-size:100%;font-family:inherit;line-height:1.15}button,input{overflow:visible}button,select{text-transform:none}button,[type=button],[type=reset],[type=submit]{-webkit-appearance:button}button::-moz-focus-inner,[type=button]::-moz-focus-inner,[type=reset]::-moz-focus-inner,[type=submit]::-moz-focus-inner{padding:0;border-style:none}button:-moz-focusring,[type=button]:-moz-focusring,[type=reset]:-moz-focusring,[type=submit]:-moz-focusring{outline:1px dotted ButtonText}fieldset{padding:.35em .75em .625em}legend{display:table;box-sizing:border-box;max-width:100%;padding:0;color:inherit;white-space:normal}progress{vertical-align:baseline}textarea{overflow:auto}[type=checkbox],[type=radio]{box-sizing:border-box;padding:0}[type=number]::-webkit-inner-spin-button,[type=number]::-webkit-outer-spin-button{height:auto}[type=search]{outline-offset:-2px;-webkit-appearance:textfield}[type=search]::-webkit-search-decoration{-webkit-appearance:none}::-webkit-file-upload-button{font:inherit;-webkit-appearance:button}details{display:block}summary{display:list-item}template{display:none}[hidden]{display:none}.arco-icon{display:inline-block;width:1em;height:1em;color:inherit;font-style:normal;vertical-align:-2px;outline:none;stroke:currentColor}.arco-icon-loading,.arco-icon-spin{animation:arco-loading-circle 1s infinite cubic-bezier(0,0,1,1)}@keyframes arco-loading-circle{0%{transform:rotate(0)}to{transform:rotate(360deg)}}.arco-icon-hover{position:relative;display:inline-block;cursor:pointer;line-height:12px}.arco-icon-hover .arco-icon{position:relative}.arco-icon-hover:before{position:absolute;display:block;box-sizing:border-box;background-color:transparent;border-radius:var(--border-radius-circle);transition:background-color .1s cubic-bezier(0,0,1,1);content:""}.arco-icon-hover:hover:before{background-color:var(--color-fill-2)}.arco-icon-hover.arco-icon-hover-disabled:before{opacity:0}.arco-icon-hover:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-mini{line-height:12px}.arco-icon-hover-size-mini:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-small{line-height:12px}.arco-icon-hover-size-small:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-large{line-height:12px}.arco-icon-hover-size-large:before{top:50%;left:50%;width:24px;height:24px;transform:translate(-50%,-50%)}.arco-icon-hover-size-huge{line-height:12px}.arco-icon-hover-size-huge:before{top:50%;left:50%;width:24px;height:24px;transform:translate(-50%,-50%)}.fade-in-standard-enter-from,.fade-in-standard-appear-from{opacity:0}.fade-in-standard-enter-to,.fade-in-standard-appear-to{opacity:1}.fade-in-standard-enter-active,.fade-in-standard-appear-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1)}.fade-in-standard-leave-from{opacity:1}.fade-in-standard-leave-to{opacity:0}.fade-in-standard-leave-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1)}.fade-in-enter-from,.fade-in-appear-from{opacity:0}.fade-in-enter-to,.fade-in-appear-to{opacity:1}.fade-in-enter-active,.fade-in-appear-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.fade-in-leave-from{opacity:1}.fade-in-leave-to{opacity:0}.fade-in-leave-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.zoom-in-enter-from,.zoom-in-appear-from{transform:scale(.5);opacity:0}.zoom-in-enter-to,.zoom-in-appear-to{transform:scale(1);opacity:1}.zoom-in-enter-active,.zoom-in-appear-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1),transform .3s cubic-bezier(.34,.69,.1,1)}.zoom-in-leave-from{transform:scale(1);opacity:1}.zoom-in-leave-to{transform:scale(.5);opacity:0}.zoom-in-leave-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1),transform .3s cubic-bezier(.34,.69,.1,1)}.zoom-in-fade-out-enter-from,.zoom-in-fade-out-appear-from{transform:scale(.5);opacity:0}.zoom-in-fade-out-enter-to,.zoom-in-fade-out-appear-to{transform:scale(1);opacity:1}.zoom-in-fade-out-enter-active,.zoom-in-fade-out-appear-active{transition:opacity .3s cubic-bezier(.3,1.3,.3,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-fade-out-leave-from{transform:scale(1);opacity:1}.zoom-in-fade-out-leave-to{transform:scale(.5);opacity:0}.zoom-in-fade-out-leave-active{transition:opacity .3s cubic-bezier(.3,1.3,.3,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-big-enter-from,.zoom-in-big-appear-from{transform:scale(.5);opacity:0}.zoom-in-big-enter-to,.zoom-in-big-appear-to{transform:scale(1);opacity:1}.zoom-in-big-enter-active,.zoom-in-big-appear-active{transition:opacity .2s cubic-bezier(0,0,1,1),transform .2s cubic-bezier(0,0,1,1)}.zoom-in-big-leave-from{transform:scale(1);opacity:1}.zoom-in-big-leave-to{transform:scale(.2);opacity:0}.zoom-in-big-leave-active{transition:opacity .2s cubic-bezier(0,0,1,1),transform .2s cubic-bezier(0,0,1,1)}.zoom-in-left-enter-from,.zoom-in-left-appear-from{transform:scale(.1);opacity:.1}.zoom-in-left-enter-to,.zoom-in-left-appear-to{transform:scale(1);opacity:1}.zoom-in-left-enter-active,.zoom-in-left-appear-active{transform-origin:0 50%;transition:opacity .3s cubic-bezier(0,0,1,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-left-leave-from{transform:scale(1);opacity:1}.zoom-in-left-leave-to{transform:scale(.1);opacity:.1}.zoom-in-left-leave-active{transform-origin:0 50%;transition:opacity .3s cubic-bezier(0,0,1,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-top-enter-from,.zoom-in-top-appear-from{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-top-enter-to,.zoom-in-top-appear-to{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-top-enter-active,.zoom-in-top-appear-active{transform-origin:0 0;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-top-leave-from{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-top-leave-to{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-top-leave-active{transform-origin:0 0;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-bottom-enter-from,.zoom-in-bottom-appear-from{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-bottom-enter-to,.zoom-in-bottom-appear-to{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-bottom-enter-active,.zoom-in-bottom-appear-active{transform-origin:100% 100%;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-bottom-leave-from{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-bottom-leave-to{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-bottom-leave-active{transform-origin:100% 100%;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.slide-dynamic-origin-enter-from,.slide-dynamic-origin-appear-from{transform:scaleY(.9);transform-origin:0 0;opacity:0}.slide-dynamic-origin-enter-to,.slide-dynamic-origin-appear-to{transform:scaleY(1);transform-origin:0 0;opacity:1}.slide-dynamic-origin-enter-active,.slide-dynamic-origin-appear-active{transition:transform .2s cubic-bezier(.34,.69,.1,1),opacity .2s cubic-bezier(.34,.69,.1,1)}.slide-dynamic-origin-leave-from{transform:scaleY(1);transform-origin:0 0;opacity:1}.slide-dynamic-origin-leave-to{transform:scaleY(.9);transform-origin:0 0;opacity:0}.slide-dynamic-origin-leave-active{transition:transform .2s cubic-bezier(.34,.69,.1,1),opacity .2s cubic-bezier(.34,.69,.1,1)}.slide-left-enter-from,.slide-left-appear-from{transform:translate(-100%)}.slide-left-enter-to,.slide-left-appear-to{transform:translate(0)}.slide-left-enter-active,.slide-left-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-left-leave-from{transform:translate(0)}.slide-left-leave-to{transform:translate(-100%)}.slide-left-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-right-enter-from,.slide-right-appear-from{transform:translate(100%)}.slide-right-enter-to,.slide-right-appear-to{transform:translate(0)}.slide-right-enter-active,.slide-right-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-right-leave-from{transform:translate(0)}.slide-right-leave-to{transform:translate(100%)}.slide-right-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-top-enter-from,.slide-top-appear-from{transform:translateY(-100%)}.slide-top-enter-to,.slide-top-appear-to{transform:translateY(0)}.slide-top-enter-active,.slide-top-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-top-leave-from{transform:translateY(0)}.slide-top-leave-to{transform:translateY(-100%)}.slide-top-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-bottom-enter-from,.slide-bottom-appear-from{transform:translateY(100%)}.slide-bottom-enter-to,.slide-bottom-appear-to{transform:translateY(0)}.slide-bottom-enter-active,.slide-bottom-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-bottom-leave-from{transform:translateY(0)}.slide-bottom-leave-to{transform:translateY(100%)}.slide-bottom-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}body{--red-1: 255,236,232;--red-2: 253,205,197;--red-3: 251,172,163;--red-4: 249,137,129;--red-5: 247,101,96;--red-6: 245,63,63;--red-7: 203,39,45;--red-8: 161,21,30;--red-9: 119,8,19;--red-10: 77,0,10;--orangered-1: 255,243,232;--orangered-2: 253,221,195;--orangered-3: 252,197,159;--orangered-4: 250,172,123;--orangered-5: 249,144,87;--orangered-6: 247,114,52;--orangered-7: 204,81,32;--orangered-8: 162,53,17;--orangered-9: 119,31,6;--orangered-10: 77,14,0;--orange-1: 255,247,232;--orange-2: 255,228,186;--orange-3: 255,207,139;--orange-4: 255,182,93;--orange-5: 255,154,46;--orange-6: 255,125,0;--orange-7: 210,95,0;--orange-8: 166,69,0;--orange-9: 121,46,0;--orange-10: 77,27,0;--gold-1: 255,252,232;--gold-2: 253,244,191;--gold-3: 252,233,150;--gold-4: 250,220,109;--gold-5: 249,204,69;--gold-6: 247,186,30;--gold-7: 204,146,19;--gold-8: 162,109,10;--gold-9: 119,75,4;--gold-10: 77,45,0;--yellow-1: 254,255,232;--yellow-2: 254,254,190;--yellow-3: 253,250,148;--yellow-4: 252,242,107;--yellow-5: 251,232,66;--yellow-6: 250,220,25;--yellow-7: 207,175,15;--yellow-8: 163,132,8;--yellow-9: 120,93,3;--yellow-10: 77,56,0;--lime-1: 252,255,232;--lime-2: 237,248,187;--lime-3: 220,241,144;--lime-4: 201,233,104;--lime-5: 181,226,65;--lime-6: 159,219,29;--lime-7: 126,183,18;--lime-8: 95,148,10;--lime-9: 67,112,4;--lime-10: 42,77,0;--green-1: 232,255,234;--green-2: 175,240,181;--green-3: 123,225,136;--green-4: 76,210,99;--green-5: 35,195,67;--green-6: 0,180,42;--green-7: 0,154,41;--green-8: 0,128,38;--green-9: 0,102,34;--green-10: 0,77,28;--cyan-1: 232,255,251;--cyan-2: 183,244,236;--cyan-3: 137,233,224;--cyan-4: 94,223,214;--cyan-5: 55,212,207;--cyan-6: 20,201,201;--cyan-7: 13,165,170;--cyan-8: 7,130,139;--cyan-9: 3,97,108;--cyan-10: 0,66,77;--blue-1: 232,247,255;--blue-2: 195,231,254;--blue-3: 159,212,253;--blue-4: 123,192,252;--blue-5: 87,169,251;--blue-6: 52,145,250;--blue-7: 32,108,207;--blue-8: 17,75,163;--blue-9: 6,48,120;--blue-10: 0,26,77;--arcoblue-1: 232,243,255;--arcoblue-2: 190,218,255;--arcoblue-3: 148,191,255;--arcoblue-4: 106,161,255;--arcoblue-5: 64,128,255;--arcoblue-6: 22,93,255;--arcoblue-7: 14,66,210;--arcoblue-8: 7,44,166;--arcoblue-9: 3,26,121;--arcoblue-10: 0,13,77;--purple-1: 245,232,255;--purple-2: 221,190,246;--purple-3: 195,150,237;--purple-4: 168,113,227;--purple-5: 141,78,218;--purple-6: 114,46,209;--purple-7: 85,29,176;--purple-8: 60,16,143;--purple-9: 39,6,110;--purple-10: 22,0,77;--pinkpurple-1: 255,232,251;--pinkpurple-2: 247,186,239;--pinkpurple-3: 240,142,230;--pinkpurple-4: 232,101,223;--pinkpurple-5: 225,62,219;--pinkpurple-6: 217,26,217;--pinkpurple-7: 176,16,182;--pinkpurple-8: 138,9,147;--pinkpurple-9: 101,3,112;--pinkpurple-10: 66,0,77;--magenta-1: 255,232,241;--magenta-2: 253,194,219;--magenta-3: 251,157,199;--magenta-4: 249,121,183;--magenta-5: 247,84,168;--magenta-6: 245,49,157;--magenta-7: 203,30,131;--magenta-8: 161,16,105;--magenta-9: 119,6,79;--magenta-10: 77,0,52;--gray-1: 247,248,250;--gray-2: 242,243,245;--gray-3: 229,230,235;--gray-4: 201,205,212;--gray-5: 169,174,184;--gray-6: 134,144,156;--gray-7: 107,119,133;--gray-8: 78,89,105;--gray-9: 39,46,59;--gray-10: 29,33,41;--success-1: var(--green-1);--success-2: var(--green-2);--success-3: var(--green-3);--success-4: var(--green-4);--success-5: var(--green-5);--success-6: var(--green-6);--success-7: var(--green-7);--success-8: var(--green-8);--success-9: var(--green-9);--success-10: var(--green-10);--primary-1: var(--arcoblue-1);--primary-2: var(--arcoblue-2);--primary-3: var(--arcoblue-3);--primary-4: var(--arcoblue-4);--primary-5: var(--arcoblue-5);--primary-6: var(--arcoblue-6);--primary-7: var(--arcoblue-7);--primary-8: var(--arcoblue-8);--primary-9: var(--arcoblue-9);--primary-10: var(--arcoblue-10);--danger-1: var(--red-1);--danger-2: var(--red-2);--danger-3: var(--red-3);--danger-4: var(--red-4);--danger-5: var(--red-5);--danger-6: var(--red-6);--danger-7: var(--red-7);--danger-8: var(--red-8);--danger-9: var(--red-9);--danger-10: var(--red-10);--warning-1: var(--orange-1);--warning-2: var(--orange-2);--warning-3: var(--orange-3);--warning-4: var(--orange-4);--warning-5: var(--orange-5);--warning-6: var(--orange-6);--warning-7: var(--orange-7);--warning-8: var(--orange-8);--warning-9: var(--orange-9);--warning-10: var(--orange-10);--link-1: var(--arcoblue-1);--link-2: var(--arcoblue-2);--link-3: var(--arcoblue-3);--link-4: var(--arcoblue-4);--link-5: var(--arcoblue-5);--link-6: var(--arcoblue-6);--link-7: var(--arcoblue-7);--link-8: var(--arcoblue-8);--link-9: var(--arcoblue-9);--link-10: var(--arcoblue-10)}body[arco-theme=dark]{--red-1: 77,0,10;--red-2: 119,6,17;--red-3: 161,22,31;--red-4: 203,46,52;--red-5: 245,78,78;--red-6: 247,105,101;--red-7: 249,141,134;--red-8: 251,176,167;--red-9: 253,209,202;--red-10: 255,240,236;--orangered-1: 77,14,0;--orangered-2: 119,30,5;--orangered-3: 162,55,20;--orangered-4: 204,87,41;--orangered-5: 247,126,69;--orangered-6: 249,146,90;--orangered-7: 250,173,125;--orangered-8: 252,198,161;--orangered-9: 253,222,197;--orangered-10: 255,244,235;--orange-1: 77,27,0;--orange-2: 121,48,4;--orange-3: 166,75,10;--orange-4: 210,105,19;--orange-5: 255,141,31;--orange-6: 255,150,38;--orange-7: 255,179,87;--orange-8: 255,205,135;--orange-9: 255,227,184;--orange-10: 255,247,232;--gold-1: 77,45,0;--gold-2: 119,75,4;--gold-3: 162,111,15;--gold-4: 204,150,31;--gold-5: 247,192,52;--gold-6: 249,204,68;--gold-7: 250,220,108;--gold-8: 252,233,149;--gold-9: 253,244,190;--gold-10: 255,252,232;--yellow-1: 77,56,0;--yellow-2: 120,94,7;--yellow-3: 163,134,20;--yellow-4: 207,179,37;--yellow-5: 250,225,60;--yellow-6: 251,233,75;--yellow-7: 252,243,116;--yellow-8: 253,250,157;--yellow-9: 254,254,198;--yellow-10: 254,255,240;--lime-1: 42,77,0;--lime-2: 68,112,6;--lime-3: 98,148,18;--lime-4: 132,183,35;--lime-5: 168,219,57;--lime-6: 184,226,75;--lime-7: 203,233,112;--lime-8: 222,241,152;--lime-9: 238,248,194;--lime-10: 253,255,238;--green-1: 0,77,28;--green-2: 4,102,37;--green-3: 10,128,45;--green-4: 18,154,55;--green-5: 29,180,64;--green-6: 39,195,70;--green-7: 80,210,102;--green-8: 126,225,139;--green-9: 178,240,183;--green-10: 235,255,236;--cyan-1: 0,66,77;--cyan-2: 6,97,108;--cyan-3: 17,131,139;--cyan-4: 31,166,170;--cyan-5: 48,201,201;--cyan-6: 63,212,207;--cyan-7: 102,223,215;--cyan-8: 144,233,225;--cyan-9: 190,244,237;--cyan-10: 240,255,252;--blue-1: 0,26,77;--blue-2: 5,47,120;--blue-3: 19,76,163;--blue-4: 41,113,207;--blue-5: 70,154,250;--blue-6: 90,170,251;--blue-7: 125,193,252;--blue-8: 161,213,253;--blue-9: 198,232,254;--blue-10: 234,248,255;--arcoblue-1: 0,13,77;--arcoblue-2: 4,27,121;--arcoblue-3: 14,50,166;--arcoblue-4: 29,77,210;--arcoblue-5: 48,111,255;--arcoblue-6: 60,126,255;--arcoblue-7: 104,159,255;--arcoblue-8: 147,190,255;--arcoblue-9: 190,218,255;--arcoblue-10: 234,244,255;--purple-1: 22,0,77;--purple-2: 39,6,110;--purple-3: 62,19,143;--purple-4: 90,37,176;--purple-5: 123,61,209;--purple-6: 142,81,218;--purple-7: 169,116,227;--purple-8: 197,154,237;--purple-9: 223,194,246;--purple-10: 247,237,255;--pinkpurple-1: 66,0,77;--pinkpurple-2: 101,3,112;--pinkpurple-3: 138,13,147;--pinkpurple-4: 176,27,182;--pinkpurple-5: 217,46,217;--pinkpurple-6: 225,61,219;--pinkpurple-7: 232,102,223;--pinkpurple-8: 240,146,230;--pinkpurple-9: 247,193,240;--pinkpurple-10: 255,242,253;--magenta-1: 77,0,52;--magenta-2: 119,8,80;--magenta-3: 161,23,108;--magenta-4: 203,43,136;--magenta-5: 245,69,166;--magenta-6: 247,86,169;--magenta-7: 249,122,184;--magenta-8: 251,158,200;--magenta-9: 253,195,219;--magenta-10: 255,232,241;--gray-1: 23,23,26;--gray-2: 46,46,48;--gray-3: 72,72,73;--gray-4: 95,95,96;--gray-5: 120,120,122;--gray-6: 146,146,147;--gray-7: 171,171,172;--gray-8: 197,197,197;--gray-9: 223,223,223;--gray-10: 246,246,246;--primary-1: var(--arcoblue-1);--primary-2: var(--arcoblue-2);--primary-3: var(--arcoblue-3);--primary-4: var(--arcoblue-4);--primary-5: var(--arcoblue-5);--primary-6: var(--arcoblue-6);--primary-7: var(--arcoblue-7);--primary-8: var(--arcoblue-8);--primary-9: var(--arcoblue-9);--primary-10: var(--arcoblue-10);--success-1: var(--green-1);--success-2: var(--green-2);--success-3: var(--green-3);--success-4: var(--green-4);--success-5: var(--green-5);--success-6: var(--green-6);--success-7: var(--green-7);--success-8: var(--green-8);--success-9: var(--green-9);--success-10: var(--green-10);--danger-1: var(--red-1);--danger-2: var(--red-2);--danger-3: var(--red-3);--danger-4: var(--red-4);--danger-5: var(--red-5);--danger-6: var(--red-6);--danger-7: var(--red-7);--danger-8: var(--red-8);--danger-9: var(--red-9);--danger-10: var(--red-10);--warning-1: var(--orange-1);--warning-2: var(--orange-2);--warning-3: var(--orange-3);--warning-4: var(--orange-4);--warning-5: var(--orange-5);--warning-6: var(--orange-6);--warning-7: var(--orange-7);--warning-8: var(--orange-8);--warning-9: var(--orange-9);--warning-10: var(--orange-10);--link-1: var(--arcoblue-1);--link-2: var(--arcoblue-2);--link-3: var(--arcoblue-3);--link-4: var(--arcoblue-4);--link-5: var(--arcoblue-5);--link-6: var(--arcoblue-6);--link-7: var(--arcoblue-7);--link-8: var(--arcoblue-8);--link-9: var(--arcoblue-9);--link-10: var(--arcoblue-10)}body{--color-white: #ffffff;--color-black: #000000;--color-border: rgb(var(--gray-3));--color-bg-popup: var(--color-bg-5);--color-bg-1: #fff;--color-bg-2: #fff;--color-bg-3: #fff;--color-bg-4: #fff;--color-bg-5: #fff;--color-bg-white: #fff;--color-neutral-1: rgb(var(--gray-1));--color-neutral-2: rgb(var(--gray-2));--color-neutral-3: rgb(var(--gray-3));--color-neutral-4: rgb(var(--gray-4));--color-neutral-5: rgb(var(--gray-5));--color-neutral-6: rgb(var(--gray-6));--color-neutral-7: rgb(var(--gray-7));--color-neutral-8: rgb(var(--gray-8));--color-neutral-9: rgb(var(--gray-9));--color-neutral-10: rgb(var(--gray-10));--color-text-1: var(--color-neutral-10);--color-text-2: var(--color-neutral-8);--color-text-3: var(--color-neutral-6);--color-text-4: var(--color-neutral-4);--color-border-1: var(--color-neutral-2);--color-border-2: var(--color-neutral-3);--color-border-3: var(--color-neutral-4);--color-border-4: var(--color-neutral-6);--color-fill-1: var(--color-neutral-1);--color-fill-2: var(--color-neutral-2);--color-fill-3: var(--color-neutral-3);--color-fill-4: var(--color-neutral-4);--color-primary-light-1: rgb(var(--primary-1));--color-primary-light-2: rgb(var(--primary-2));--color-primary-light-3: rgb(var(--primary-3));--color-primary-light-4: rgb(var(--primary-4));--color-link-light-1: rgb(var(--link-1));--color-link-light-2: rgb(var(--link-2));--color-link-light-3: rgb(var(--link-3));--color-link-light-4: rgb(var(--link-4));--color-secondary: var(--color-neutral-2);--color-secondary-hover: var(--color-neutral-3);--color-secondary-active: var(--color-neutral-4);--color-secondary-disabled: var(--color-neutral-1);--color-danger-light-1: rgb(var(--danger-1));--color-danger-light-2: rgb(var(--danger-2));--color-danger-light-3: rgb(var(--danger-3));--color-danger-light-4: rgb(var(--danger-4));--color-success-light-1: rgb(var(--success-1));--color-success-light-2: rgb(var(--success-2));--color-success-light-3: rgb(var(--success-3));--color-success-light-4: rgb(var(--success-4));--color-warning-light-1: rgb(var(--warning-1));--color-warning-light-2: rgb(var(--warning-2));--color-warning-light-3: rgb(var(--warning-3));--color-warning-light-4: rgb(var(--warning-4));--border-radius-none: 0;--border-radius-small: 2px;--border-radius-medium: 4px;--border-radius-large: 8px;--border-radius-circle: 50%;--color-tooltip-bg: rgb(var(--gray-10));--color-spin-layer-bg: rgba(255, 255, 255, .6);--color-menu-dark-bg: #232324;--color-menu-light-bg: #ffffff;--color-menu-dark-hover: rgba(255, 255, 255, .04);--color-mask-bg: rgba(29, 33, 41, .6)}body[arco-theme=dark]{--color-white: rgba(255, 255, 255, .9);--color-black: #000000;--color-border: #333335;--color-bg-1: #17171a;--color-bg-2: #232324;--color-bg-3: #2a2a2b;--color-bg-4: #313132;--color-bg-5: #373739;--color-bg-white: #f6f6f6;--color-text-1: rgba(255, 255, 255, .9);--color-text-2: rgba(255, 255, 255, .7);--color-text-3: rgba(255, 255, 255, .5);--color-text-4: rgba(255, 255, 255, .3);--color-fill-1: rgba(255, 255, 255, .04);--color-fill-2: rgba(255, 255, 255, .08);--color-fill-3: rgba(255, 255, 255, .12);--color-fill-4: rgba(255, 255, 255, .16);--color-primary-light-1: rgba(var(--primary-6), .2);--color-primary-light-2: rgba(var(--primary-6), .35);--color-primary-light-3: rgba(var(--primary-6), .5);--color-primary-light-4: rgba(var(--primary-6), .65);--color-secondary: rgba(var(--gray-9), .08);--color-secondary-hover: rgba(var(--gray-8), .16);--color-secondary-active: rgba(var(--gray-7), .24);--color-secondary-disabled: rgba(var(--gray-9), .08);--color-danger-light-1: rgba(var(--danger-6), .2);--color-danger-light-2: rgba(var(--danger-6), .35);--color-danger-light-3: rgba(var(--danger-6), .5);--color-danger-light-4: rgba(var(--danger-6), .65);--color-success-light-1: rgb(var(--success-6), .2);--color-success-light-2: rgb(var(--success-6), .35);--color-success-light-3: rgb(var(--success-6), .5);--color-success-light-4: rgb(var(--success-6), .65);--color-warning-light-1: rgb(var(--warning-6), .2);--color-warning-light-2: rgb(var(--warning-6), .35);--color-warning-light-3: rgb(var(--warning-6), .5);--color-warning-light-4: rgb(var(--warning-6), .65);--color-link-light-1: rgb(var(--link-6), .2);--color-link-light-2: rgb(var(--link-6), .35);--color-link-light-3: rgb(var(--link-6), .5);--color-link-light-4: rgb(var(--link-6), .65);--color-tooltip-bg: #373739;--color-spin-layer-bg: rgba(51, 51, 51, .6);--color-menu-dark-bg: #232324;--color-menu-light-bg: #232324;--color-menu-dark-hover: var(--color-fill-2);--color-mask-bg: rgba(23, 23, 26, .6)}body{font-size:14px;font-family:Inter,-apple-system,BlinkMacSystemFont,PingFang SC,Hiragino Sans GB,noto sans,Microsoft YaHei,Helvetica Neue,Helvetica,Arial,sans-serif}.arco-message-list{position:fixed;z-index:1003;display:flex;flex-direction:column;align-items:center;box-sizing:border-box;width:100%;margin:0;padding:0 10px;text-align:center;pointer-events:none}.arco-message-list-top{top:40px}.arco-message-list-bottom{bottom:40px}.arco-message{position:relative;display:inline-flex;align-items:center;margin-bottom:16px;padding:10px 16px;overflow:hidden;line-height:1;text-align:center;list-style:none;background-color:var(--color-bg-popup);border:1px solid var(--color-neutral-3);border-radius:var(--border-radius-small);box-shadow:0 4px 10px #0000001a;transition:all .1s cubic-bezier(0,0,1,1);pointer-events:auto}.arco-message-icon{display:inline-block;margin-right:8px;color:var(--color-text-1);font-size:20px;vertical-align:middle;animation:arco-msg-fade .1s cubic-bezier(0,0,1,1),arco-msg-fade .4s cubic-bezier(.3,1.3,.3,1)}.arco-message-content{font-size:14px;color:var(--color-text-1);vertical-align:middle}.arco-message-info{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-info .arco-message-icon{color:rgb(var(--primary-6))}.arco-message-info .arco-message-content{color:var(--color-text-1)}.arco-message-success{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-success .arco-message-icon{color:rgb(var(--success-6))}.arco-message-success .arco-message-content{color:var(--color-text-1)}.arco-message-warning{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-warning .arco-message-icon{color:rgb(var(--warning-6))}.arco-message-warning .arco-message-content{color:var(--color-text-1)}.arco-message-error{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-error .arco-message-icon{color:rgb(var(--danger-6))}.arco-message-error .arco-message-content{color:var(--color-text-1)}.arco-message-loading{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-loading .arco-message-icon{color:rgb(var(--primary-6))}.arco-message-loading .arco-message-content{color:var(--color-text-1)}.arco-message-close-btn{margin-left:8px;color:var(--color-text-1);font-size:12px}.arco-message .arco-icon-hover.arco-message-icon-hover:before{width:20px;height:20px}.fade-message-enter-from,.fade-message-appear-from{opacity:0}.fade-message-enter-to,.fade-message-appear-to{opacity:1}.fade-message-enter-active,.fade-message-appear-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.fade-message-leave-from{opacity:1}.fade-message-leave-to{opacity:0}.fade-message-leave-active{position:absolute}.flip-list-move{transition:transform .8s ease}@keyframes arco-msg-fade{0%{opacity:0}to{opacity:1}}@keyframes arco-msg-scale{0%{transform:scale(0)}to{transform:scale(1)}}:root{--c-brand: #3eaf7c;--c-brand-light: #4abf8a;--c-bg: #ffffff;--c-bg-light: #f3f4f5;--c-bg-lighter: #eeeeee;--c-bg-navbar: var(--c-bg);--c-bg-sidebar: var(--c-bg);--c-bg-arrow: #cccccc;--c-text: #2c3e50;--c-text-accent: var(--c-brand);--c-text-light: #3a5169;--c-text-lighter: #4e6e8e;--c-text-lightest: #6a8bad;--c-text-quote: #999999;--c-border: #eaecef;--c-border-dark: #dfe2e5;--c-tip: #42b983;--c-tip-bg: var(--c-bg-light);--c-tip-title: var(--c-text);--c-tip-text: var(--c-text);--c-tip-text-accent: var(--c-text-accent);--c-warning: #e7c000;--c-warning-bg: #fffae3;--c-warning-title: #ad9000;--c-warning-text: #746000;--c-warning-text-accent: var(--c-text);--c-danger: #cc0000;--c-danger-bg: #ffe0e0;--c-danger-title: #990000;--c-danger-text: #660000;--c-danger-text-accent: var(--c-text);--c-details-bg: #eeeeee;--c-badge-tip: var(--c-tip);--c-badge-warning: var(--c-warning);--c-badge-danger: var(--c-danger);--t-color: .3s ease;--t-transform: .3s ease;--code-bg-color: #282c34;--code-hl-bg-color: rgba(0, 0, 0, .66);--code-ln-color: #9e9e9e;--code-ln-wrapper-width: 3.5rem;--font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif;--font-family-code: Consolas, Monaco, "Andale Mono", "Ubuntu Mono", monospace;--navbar-height: 3.6rem;--navbar-padding-v: .7rem;--navbar-padding-h: 1.5rem;--sidebar-width: 20rem;--sidebar-width-mobile: calc(var(--sidebar-width) * .82);--content-width: 740px;--homepage-width: 960px}html.dark{--c-brand: #3aa675;--c-brand-light: #349469;--c-bg: #22272e;--c-bg-light: #2b313a;--c-bg-lighter: #262c34;--c-text: #adbac7;--c-text-light: #96a7b7;--c-text-lighter: #8b9eb0;--c-text-lightest: #8094a8;--c-border: #3e4c5a;--c-border-dark: #34404c;--c-tip: #318a62;--c-warning: #ceab00;--c-warning-bg: #7e755b;--c-warning-title: #ceac03;--c-warning-text: #362e00;--c-danger: #940000;--c-danger-bg: #806161;--c-danger-title: #610000;--c-danger-text: #3a0000;--c-details-bg: #323843;--code-hl-bg-color: #363b46}@font-face{font-family:octicons-link;src:url(data:font/woff;charset=utf-8;base64,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) format("woff")}.markdown-body{-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%;color:var(--c-text);font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Helvetica,Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji",Segoe UI Symbol;font-size:14px;line-height:1.5;word-wrap:break-word;transition:background-color var(--t-color),border-color var(--t-color)}.g-emoji{display:inline-block;min-width:1ch;font-family:"Apple Color Emoji","Segoe UI Emoji",Segoe UI Symbol,serif;font-size:1em;font-style:normal!important;font-weight:400;line-height:1;vertical-align:-.075em}.markdown-body .pl-c{color:#6a737d}.markdown-body .pl-c1,.markdown-body .pl-s .pl-v{color:#005cc5}.markdown-body .pl-e,.markdown-body .pl-en{color:#6f42c1}.markdown-body .pl-smi,.markdown-body .pl-s .pl-s1{color:var(--c-text)}.markdown-body .pl-ent{color:#22863a}.markdown-body .pl-k{color:#d73a49}.markdown-body .pl-s,.markdown-body .pl-pds,.markdown-body .pl-s .pl-pse .pl-s1,.markdown-body .pl-sr,.markdown-body .pl-sr .pl-cce,.markdown-body .pl-sr .pl-sre,.markdown-body .pl-sr .pl-sra{color:var(--c-text-lighter)}.markdown-body .pl-v,.markdown-body .pl-smw{color:#e36209}.markdown-body .pl-bu{color:#b31d28}.markdown-body .pl-ii{color:#fafbfc;background-color:#b31d28}.markdown-body .pl-c2{color:#fafbfc;background-color:#d73a49}.markdown-body .pl-c2:before{content:"^M"}.markdown-body .pl-sr .pl-cce{font-weight:700;color:#22863a}.markdown-body .pl-ml{color:#735c0f}.markdown-body .pl-mh,.markdown-body .pl-mh .pl-en,.markdown-body .pl-ms{font-weight:700;color:#005cc5}.markdown-body .pl-mi{font-style:italic;color:#24292e}.markdown-body .pl-mb{font-weight:700;color:#24292e}.markdown-body .pl-md{color:#b31d28;background-color:#ffeef0}.markdown-body .pl-mi1{color:#22863a;background-color:#f0fff4}.markdown-body .pl-mc{color:#e36209;background-color:#ffebda}.markdown-body .pl-mi2{color:#f6f8fa;background-color:#005cc5}.markdown-body .pl-mdr{font-weight:700;color:#6f42c1}.markdown-body .pl-ba{color:#586069}.markdown-body .pl-sg{color:#959da5}.markdown-body .pl-corl{text-decoration:underline;color:var(--c-text-lighter)}.markdown-body .octicon{display:inline-block;vertical-align:text-top;fill:currentColor}.markdown-body a{background-color:transparent;-webkit-text-decoration-skip:objects}.markdown-body a:active,.markdown-body a:hover{outline-width:0}.markdown-body strong{font-weight:inherit}.markdown-body strong{font-weight:bolder}.markdown-body h1{font-size:2em;margin:.67em 0}.markdown-body img{border-style:none}.markdown-body svg:not(:root){overflow:hidden}.markdown-body code,.markdown-body kbd,.markdown-body pre{font-family:monospace,monospace;font-size:1em}.markdown-body hr{-webkit-box-sizing:content-box;box-sizing:content-box;height:0;overflow:visible}.markdown-body input{font:inherit;margin:0}.markdown-body input{overflow:visible}.markdown-body [type=checkbox]{-webkit-box-sizing:border-box;box-sizing:border-box;padding:0}.markdown-body *{-webkit-box-sizing:border-box;box-sizing:border-box}.markdown-body input{font-family:inherit;font-size:inherit;line-height:inherit}.markdown-body a{color:#0366d6;text-decoration:none}.markdown-body a:hover{text-decoration:underline}.markdown-body strong{font-weight:600}.markdown-body hr{height:0;margin:15px 0;overflow:hidden;background:transparent;border:0;border-bottom:1px solid #dfe2e5}.markdown-body hr:before{display:table;content:""}.markdown-body hr:after{display:table;clear:both;content:""}.markdown-body table{border-spacing:0;border-collapse:collapse}.markdown-body td,.markdown-body th{padding:0}.markdown-body h1,.markdown-body h2,.markdown-body h3,.markdown-body h4,.markdown-body h5,.markdown-body h6{margin-top:0;margin-bottom:0}.markdown-body h1{font-size:32px;font-weight:600}.markdown-body h2{font-size:24px;font-weight:600}.markdown-body h3{font-size:20px;font-weight:600}.markdown-body h4,.markdown-body h5{font-size:14px;font-weight:600}.markdown-body h6{font-size:12px;font-weight:600}.markdown-body p{margin-top:0;margin-bottom:10px}.markdown-body blockquote{margin:0}.markdown-body ul,.markdown-body ol{padding-left:0;margin-top:0;margin-bottom:0}.markdown-body ol ol,.markdown-body ul ol{list-style-type:lower-roman}.markdown-body ul ul ol,.markdown-body ul ol ol,.markdown-body ol ul ol,.markdown-body ol ol ol{list-style-type:lower-alpha}.markdown-body dd{margin-left:0}.markdown-body code{font-family:SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace;font-size:12px}.markdown-body pre{margin-top:0;margin-bottom:0;font:12px SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace}.markdown-body .octicon{vertical-align:text-bottom}.markdown-body .pl-0{padding-left:0!important}.markdown-body .pl-1{padding-left:4px!important}.markdown-body .pl-2{padding-left:8px!important}.markdown-body .pl-3{padding-left:14px!important}.markdown-body .pl-4{padding-left:24px!important}.markdown-body .pl-5{padding-left:32px!important}.markdown-body .pl-6{padding-left:40px!important}.markdown-body:before{display:table;content:""}.markdown-body:after{display:table;clear:both;content:""}.markdown-body>*:first-child{margin-top:0!important}.markdown-body>*:last-child{margin-bottom:0!important}.markdown-body a:not([href]){color:inherit;text-decoration:none}.markdown-body .anchor{float:left;padding-right:4px;margin-left:-20px;line-height:1}.markdown-body .anchor:focus{outline:none}.markdown-body p,.markdown-body blockquote,.markdown-body ul,.markdown-body ol,.markdown-body dl,.markdown-body table,.markdown-body pre{margin-top:0;margin-bottom:14px}.markdown-body hr{height:.25em;padding:0;margin:24px 0;background-color:#e1e4e8;border:0}.markdown-body blockquote{padding:0 1em;color:#6a737d;border-left:.25em solid var(--c-border)}.markdown-body blockquote>:first-child{margin-top:0}.markdown-body blockquote>:last-child{margin-bottom:0}.markdown-body kbd{display:inline-block;padding:3px 5px;font-size:11px;line-height:10px;color:#444d56;vertical-align:middle;background-color:#fafbfc;border:solid 1px #c6cbd1;border-bottom-color:#959da5;border-radius:3px;-webkit-box-shadow:inset 0 -1px 0 #959da5;box-shadow:inset 0 -1px #959da5}.markdown-body h1,.markdown-body h2,.markdown-body h3,.markdown-body h4,.markdown-body h5,.markdown-body h6{margin-top:24px;margin-bottom:14px;font-weight:600;line-height:1.25}.markdown-body h1 .octicon-link,.markdown-body h2 .octicon-link,.markdown-body h3 .octicon-link,.markdown-body h4 .octicon-link,.markdown-body h5 .octicon-link,.markdown-body h6 .octicon-link{color:#1b1f23;vertical-align:middle;visibility:hidden}.markdown-body h1:hover .anchor,.markdown-body h2:hover .anchor,.markdown-body h3:hover .anchor,.markdown-body h4:hover .anchor,.markdown-body h5:hover .anchor,.markdown-body h6:hover .anchor{text-decoration:none}.markdown-body h1:hover .anchor .octicon-link,.markdown-body h2:hover .anchor .octicon-link,.markdown-body h3:hover .anchor .octicon-link,.markdown-body h4:hover .anchor .octicon-link,.markdown-body h5:hover .anchor .octicon-link,.markdown-body h6:hover .anchor .octicon-link{visibility:visible}.markdown-body h1{padding-bottom:.3em;font-size:2em;border-bottom:1px solid #eaecef}.markdown-body h2{padding-bottom:.3em;font-size:1.5em;border-bottom:1px solid #eaecef}.markdown-body h3{font-size:1.25em}.markdown-body h4{font-size:1em}.markdown-body h5{font-size:.875em}.markdown-body h6{font-size:.85em;color:#6a737d}.markdown-body ul,.markdown-body ol{padding-left:2em}.markdown-body ul ul,.markdown-body ul ol,.markdown-body ol ol,.markdown-body ol ul{margin-top:0;margin-bottom:0}.markdown-body li>p{margin-top:14px}.markdown-body li+li{margin-top:.25em}.markdown-body dl{padding:0}.markdown-body dl dt{padding:0;margin-top:14px;font-size:1em;font-style:italic;font-weight:600}.markdown-body dl dd{padding:0 14px;margin-bottom:14px}.markdown-body table{display:block;width:100%;overflow:auto}.markdown-body table th{font-weight:600}.markdown-body table th,.markdown-body table td{padding:6px 13px;border:1px solid #dfe2e5}.markdown-body table tr{background-color:#fff;border-top:1px solid #c6cbd1}.markdown-body table tr:nth-child(2n){background-color:#f6f8fa}.markdown-body img{max-width:100%;-webkit-box-sizing:content-box;box-sizing:content-box;background-color:#fff}.markdown-body code{padding:.2em 0;margin:0;font-size:85%;border-radius:3px}.markdown-body code:before,.markdown-body code:after{letter-spacing:-.2em;content:" "}.markdown-body pre{word-wrap:normal}.markdown-body pre>code{padding:0;margin:0;font-size:100%;word-break:normal;white-space:pre;background:transparent;border:0}.markdown-body .highlight{margin-bottom:14px}.markdown-body .highlight pre{margin-bottom:0;word-break:normal}.markdown-body .highlight pre,.markdown-body pre{padding:14px;overflow:auto;font-size:85%;line-height:1.45;background-color:var(--c-tip-bg);border-radius:3px;color:var(--c-text);transition:background-color var(--t-color),border-color var(--t-color)}.markdown-body pre code{display:inline;max-width:auto;padding:0;margin:0;overflow:visible;line-height:inherit;word-wrap:normal;background-color:transparent;border:0;color:var(--c-text)}.markdown-body pre code:before,.markdown-body pre code:after{content:normal}.markdown-body .full-commit .btn-outline:not(:disabled):hover{color:#005cc5;border-color:#005cc5}.markdown-body kbd{display:inline-block;padding:3px 5px;font:11px SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace;line-height:10px;color:#444d56;vertical-align:middle;background-color:#fafbfc;border:solid 1px #d1d5da;border-bottom-color:#c6cbd1;border-radius:3px;-webkit-box-shadow:inset 0 -1px 0 #c6cbd1;box-shadow:inset 0 -1px #c6cbd1}.markdown-body :checked+.radio-label{position:relative;z-index:1;border-color:#0366d6}.markdown-body .task-list-item{list-style-type:none}.markdown-body .task-list-item+.task-list-item{margin-top:3px}.markdown-body .task-list-item input{margin:0 .2em .25em -1.6em;vertical-align:middle}.markdown-body hr{border-bottom-color:#eee}.gt-container{-webkit-box-sizing:border-box;box-sizing:border-box;font-size:14px}.gt-container *{-webkit-box-sizing:border-box;box-sizing:border-box}.gt-container a{color:var(--c-text-accent);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container a.is--active{color:var(--c-text);cursor:default!important}.gt-container a.is--active:hover{color:var(--c-text-accent)}.gt-container .hide{display:none!important}.gt-container .gt-svg{display:inline-block;width:1em;height:1em;vertical-align:sub}.gt-container .gt-ico-arrdown{fill:var(--c-text)}.gt-container .gt-svg svg{width:100%;height:100%;fill:var(--c-text-accent)}.gt-container .gt-ico{display:inline-block}.gt-container .gt-ico-text{margin-left:.3125em}.gt-container .gt-ico-tip{font-size:.875em}.gt-container .gt-ico-github,.gt-container .gt-ico-github .gt-svg{width:100%;height:100%}.gt-container .gt-ico-github svg{fill:var(--c-text)}.gt-container .gt-spinner{position:relative}.gt-container .gt-spinner:before{content:"";-webkit-box-sizing:border-box;box-sizing:border-box;position:absolute;top:3px;width:.75em;height:.75em;margin-top:-.1875em;margin-left:-.375em;border-radius:50%;border:1px solid #fff;border-top-color:var(--c-text-accent);-webkit-animation:gt-kf-rotate .6s linear infinite;animation:gt-kf-rotate .6s linear infinite}.gt-container .gt-loader{position:relative;border:1px solid #999;-webkit-animation:ease gt-kf-rotate 1.5s infinite;animation:ease gt-kf-rotate 1.5s infinite;display:inline-block;font-style:normal;width:1.75em;height:1.75em;line-height:1.75em;border-radius:50%}.gt-container .gt-loader:before{content:"";position:absolute;display:block;top:0;left:50%;margin-top:-.1875em;margin-left:-.1875em;width:.375em;height:.375em;background-color:#999;border-radius:50%}.gt-container .gt-avatar{display:inline-block;width:3.125em;height:3.125em}@media (max-width: 479px){.gt-container .gt-avatar{width:2em;height:2em}}.gt-container .gt-avatar img{width:100%;height:auto;border-radius:3px}.gt-container .gt-avatar-github{width:3.125em;height:3.125em;cursor:pointer}@media (max-width: 479px){.gt-container .gt-avatar-github{width:2em;height:2em}}.gt-container .gt-btn{padding:.75em .95em;display:inline-block;line-height:1;text-decoration:none;white-space:nowrap;cursor:pointer;border:1px solid var(--c-text-accent);border-radius:5px;background-color:var(--c-brand);color:#fff;outline:none;font-size:.75em;transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-btn-text{font-weight:400}.gt-header-controls .gt-btn-public .gt-btn-text{color:var(--c-bg)}.gt-container .gt-btn-loading{position:relative;margin-left:.5em;display:inline-block;width:.75em;height:1em;vertical-align:top}.gt-container .gt-btn.is--disable{cursor:not-allowed;opacity:.5}.gt-container .gt-btn-login{margin-right:0;color:var(--c-bg)}.gt-container .gt-btn-login:hover{background-color:var(--c-brand-light);border-color:var(--c-brand)}.gt-container .gt-btn-preview{background-color:var(--c-bg);color:var(--c-brand);border-color:var(--c-brand);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-btn-preview:hover{color:var(--c-bg);background-color:var(--c-brand-light)}.gt-container .gt-btn-public:hover,.gt-container .gt-btn-loadmore:hover{background-color:var(--c-brand-light);border-color:var(--c-brand)}.gt-container .gt-btn-loadmore{color:var(--c-bg)}.gt-container .gt-error{text-align:center;margin:.625em;color:#ff3860}.gt-container .gt-initing{padding:1.25em 0;text-align:center}.gt-container .gt-initing-text{margin:.625em auto;font-size:92%}.gt-container .gt-no-init{padding:1.25em 0;text-align:center}.gt-container .gt-link{border-bottom:1px dotted var(--c-text-accent)}.gt-container .gt-link-counts,.gt-container .gt-link-project{text-decoration:none}.gt-container .gt-meta{margin:1.25em 0;padding:1em 0;position:relative;border-bottom:1px solid var(--c-border);font-size:1em;transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-meta:before,.gt-container .gt-meta:after{content:" ";display:table}.gt-container .gt-meta:after{clear:both}.gt-container .gt-counts{margin:0 .625em 0 0}.gt-container .gt-user{float:right;margin:0;font-size:92%}.gt-container .gt-user-pic{width:14px;height:14px;vertical-align:top;margin-right:.5em}.gt-container .gt-user-inner{display:inline-block;cursor:pointer}.gt-container .gt-user .gt-ico{margin:0 0 0 .3125em}.gt-container .gt-user .gt-ico svg{fill:inherit}.gt-container .gt-user .is--poping .gt-ico svg{fill:var(--c-text-accent)}.gt-container .gt-version{color:#a1a1a1;margin-left:.375em}.gt-container .gt-copyright{margin:0 .9375em .5em;border-top:1px solid var(--c-border);padding-top:.5em}.gt-container .gt-popup{position:absolute;right:0;top:2.375em;background-color:var(--c-bg-navbar);display:block!important;border:1px solid var(--c-border);padding:.625em 0;font-size:.875em;letter-spacing:.5px;box-sizing:border-box;z-index:10}.gt-container .gt-popup .gt-action{cursor:pointer;display:block;margin:.5em 0;padding:0 1.125em;position:relative;text-decoration:none}.gt-container .gt-popup .gt-action.is--active:before{content:"";width:.25em;height:.25em;background:var(--c-text-accent);position:absolute;left:.5em;top:.4375em}.gt-container .gt-header{position:relative;display:-webkit-box;display:-ms-flexbox;display:flex}.gt-container .gt-header-comment{-webkit-box-flex:1;-ms-flex:1;flex:1;padding-left:1.25em;width:90%}@media (max-width: 479px){.gt-container .gt-header-comment{padding-left:.875em}}.gt-container .gt-header-textarea{padding:.75em;display:block;-webkit-box-sizing:border-box;box-sizing:border-box;width:100%;min-height:5.125em;max-height:15em;border-radius:5px;border:1px solid var(--c-border);font-size:.875em;word-wrap:break-word;resize:vertical;background-color:var(--c-bg);outline:none;-webkit-transition:all .25s ease;transition:background-color var(--t-color),border-color var(--t-color)}.gt-header-textarea{color:var(--c-text)}.gt-container .gt-header-preview{padding:.75em;border-radius:5px;border:1px solid var(--c-border);background-color:var(--c-bg)}.gt-container .gt-header-controls{position:relative;margin:.75em 0 0}.gt-container .gt-header-controls:before,.gt-container .gt-header-controls:after{content:" ";display:table}.gt-container .gt-header-controls:after{clear:both}@media (max-width: 479px){.gt-container .gt-header-controls{margin:0}}.gt-container .gt-header-controls-tip{font-size:13px;color:var(--c-text-accent);text-decoration:none;vertical-align:sub}@media (max-width: 479px){.gt-container .gt-header-controls-tip{display:none}}.gt-container .gt-header-controls .gt-btn{float:right;margin-left:1.2em}@media (max-width: 479px){.gt-container .gt-header-controls .gt-btn{float:none;width:100%;margin:.75em 0 0}}.gt-container:after{content:"";position:fixed;bottom:100%;left:0;right:0;top:0;opacity:0}.gt-container.gt-input-focused{position:relative}.gt-container.gt-input-focused:after{content:"";position:fixed;bottom:0%;left:0;right:0;top:0;background:#000;opacity:.6;-webkit-transition:opacity .3s,bottom 0s;transition:opacity .3s,bottom 0s;z-index:9999}.gt-container.gt-input-focused .gt-header-comment{z-index:10000}.gt-container .gt-comments{padding-top:1.25em}.gt-container .gt-comments-null{text-align:center}.gt-container .gt-comments-controls{margin:1.25em 0;text-align:center}.gt-container .gt-comment{position:relative;padding:.625em 0;display:-webkit-box;display:-ms-flexbox;display:flex}.gt-container .gt-comment-content{-webkit-box-flex:1;-ms-flex:1;flex:1;margin-left:1.25em;padding:.75em 1em;background-color:var(--c-bg);overflow:auto;-webkit-transition:all ease .25s;border:1px solid var(--c-border);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-comment-content{border-radius:5px}@media (max-width: 479px){.gt-container .gt-comment-content{margin-left:.875em;padding:.625em .75em}}.gt-container .gt-comment-header{margin-bottom:.75em;font-size:.875em;position:relative}.gt-container .gt-comment-block-1{float:right;height:1.375em;width:3.5em}.gt-container .gt-comment-block-2{float:right;height:1.375em;width:4em}.gt-container .gt-comment-username{font-weight:500;color:var(--c-text-accent);text-decoration:none}.gt-container .gt-comment-username:hover{text-decoration:underline}.gt-container .gt-comment-text,.gt-container .gt-comment-date{margin-left:.5em;color:#a1a1a1;font-size:.875em}.gt-container .gt-comment-like,.gt-container .gt-comment-edit,.gt-container .gt-comment-reply{position:absolute;height:1.375em}.gt-container .gt-comment-like:hover,.gt-container .gt-comment-edit:hover,.gt-container .gt-comment-reply:hover{cursor:pointer}.gt-container .gt-comment-like{top:0;right:2em}.gt-container .gt-comment-edit,.gt-container .gt-comment-reply{top:0;right:0}.gt-container .gt-comment-body{color:var(--c-text)}.gt-container .gt-comment-body .email-hidden-toggle a{display:inline-block;height:12px;padding:0 9px;font-size:12px;font-weight:600;line-height:6px;color:#444d56;text-decoration:none;vertical-align:middle;background:#dfe2e5;border-radius:1px}.gt-container .gt-comment-body .email-hidden-toggle a:hover{background-color:#c6cbd1}.gt-container .gt-comment-body .email-hidden-reply{display:none;white-space:pre-wrap}.gt-container .gt-comment-body .email-hidden-reply .email-signature-reply{padding:0 15px;margin:15px 0;color:#586069;border-left:4px solid #dfe2e5}.gt-container .gt-comment-body .email-hidden-reply.expanded{display:block}.gt-container .gt-comment-admin .gt-comment-content{background-color:var(--c-bg);border:1px solid var(--c-border)}@-webkit-keyframes gt-kf-rotate{0%{-webkit-transform:rotate(0);transform:rotate(0)}to{-webkit-transform:rotate(360deg);transform:rotate(360deg)}}@keyframes gt-kf-rotate{0%{-webkit-transform:rotate(0);transform:rotate(0)}to{-webkit-transform:rotate(360deg);transform:rotate(360deg)}}.gt-comment-body .snippet-clipboard-content pre code{color:var(--c-text)}
+@charset "UTF-8";@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-cyrillic.5f2c6c8c.woff2) format("woff2");unicode-range:U+0301,U+0400-045F,U+0490-0491,U+04B0-04B1,U+2116}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-cyrillic-ext.e75737ce.woff2) format("woff2");unicode-range:U+0460-052F,U+1C80-1C88,U+20B4,U+2DE0-2DFF,U+A640-A69F,U+FE2E-FE2F}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-greek.d5a6d92a.woff2) format("woff2");unicode-range:U+0370-03FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-greek-ext.ab0619bc.woff2) format("woff2");unicode-range:U+1F00-1FFF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-latin.2ed14f66.woff2) format("woff2");unicode-range:U+0000-00FF,U+0131,U+0152-0153,U+02BB-02BC,U+02C6,U+02DA,U+02DC,U+2000-206F,U+2074,U+20AC,U+2122,U+2191,U+2193,U+2212,U+2215,U+FEFF,U+FFFD}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-latin-ext.0030eebd.woff2) format("woff2");unicode-range:U+0100-024F,U+0259,U+1E00-1EFF,U+2020,U+20A0-20AB,U+20AD-20CF,U+2113,U+2C60-2C7F,U+A720-A7FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:normal;font-named-instance:"Regular";src:url(/assets/inter-roman-vietnamese.14ce25a6.woff2) format("woff2");unicode-range:U+0102-0103,U+0110-0111,U+0128-0129,U+0168-0169,U+01A0-01A1,U+01AF-01B0,U+1EA0-1EF9,U+20AB}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-cyrillic.ea42a392.woff2) format("woff2");unicode-range:U+0301,U+0400-045F,U+0490-0491,U+04B0-04B1,U+2116}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-cyrillic-ext.33bd5a8e.woff2) format("woff2");unicode-range:U+0460-052F,U+1C80-1C88,U+20B4,U+2DE0-2DFF,U+A640-A69F,U+FE2E-FE2F}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-greek.8f4463c4.woff2) format("woff2");unicode-range:U+0370-03FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-greek-ext.4fbe9427.woff2) format("woff2");unicode-range:U+1F00-1FFF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-latin.bd3b6f56.woff2) format("woff2");unicode-range:U+0000-00FF,U+0131,U+0152-0153,U+02BB-02BC,U+02C6,U+02DA,U+02DC,U+2000-206F,U+2074,U+20AC,U+2122,U+2191,U+2193,U+2212,U+2215,U+FEFF,U+FFFD}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-latin-ext.bd8920cc.woff2) format("woff2");unicode-range:U+0100-024F,U+0259,U+1E00-1EFF,U+2020,U+20A0-20AB,U+20AD-20CF,U+2113,U+2C60-2C7F,U+A720-A7FF}@font-face{font-family:Inter var;font-weight:100 900;font-display:swap;font-style:italic;font-named-instance:"Italic";src:url(/assets/inter-italic-vietnamese.6ce511fb.woff2) format("woff2");unicode-range:U+0102-0103,U+0110-0111,U+0128-0129,U+0168-0169,U+01A0-01A1,U+01AF-01B0,U+1EA0-1EF9,U+20AB}@font-face{font-family:Chinese Quotes;src:local("PingFang SC Regular"),local("PingFang SC"),local("SimHei"),local("Source Han Sans SC");unicode-range:U+2018,U+2019,U+201C,U+201D}:root{--vp-c-white: #ffffff;--vp-c-black: #000000;--vp-c-gray: #8e8e93;--vp-c-text-light-1: rgba(60, 60, 67);--vp-c-text-light-2: rgba(60, 60, 67, .75);--vp-c-text-light-3: rgba(60, 60, 67, .33);--vp-c-text-dark-1: rgba(255, 255, 245, .86);--vp-c-text-dark-2: rgba(235, 235, 245, .6);--vp-c-text-dark-3: rgba(235, 235, 245, .38);--vp-c-green: #10b981;--vp-c-green-light: #34d399;--vp-c-green-lighter: #6ee7b7;--vp-c-green-dark: #059669;--vp-c-green-darker: #047857;--vp-c-green-dimm-1: rgba(16, 185, 129, .05);--vp-c-green-dimm-2: rgba(16, 185, 129, .2);--vp-c-green-dimm-3: rgba(16, 185, 129, .5);--vp-c-yellow: #d97706;--vp-c-yellow-light: #f59e0b;--vp-c-yellow-lighter: #fbbf24;--vp-c-yellow-dark: #b45309;--vp-c-yellow-darker: #92400e;--vp-c-yellow-dimm-1: rgba(234, 179, 8, .05);--vp-c-yellow-dimm-2: rgba(234, 179, 8, .2);--vp-c-yellow-dimm-3: rgba(234, 179, 8, .5);--vp-c-red: #f43f5e;--vp-c-red-light: #fb7185;--vp-c-red-lighter: #fda4af;--vp-c-red-dark: #e11d48;--vp-c-red-darker: #be123c;--vp-c-red-dimm-1: rgba(244, 63, 94, .05);--vp-c-red-dimm-2: rgba(244, 63, 94, .2);--vp-c-red-dimm-3: rgba(244, 63, 94, .5);--vp-c-sponsor: #db2777}:root{--vp-c-bg: #ffffff;--vp-c-bg-elv: #ffffff;--vp-c-bg-elv-up: #ffffff;--vp-c-bg-elv-down: #f6f6f7;--vp-c-bg-elv-mute: #f6f6f7;--vp-c-bg-soft: #f6f6f7;--vp-c-bg-soft-up: #f9f9fa;--vp-c-bg-soft-down: #e3e3e5;--vp-c-bg-soft-mute: #e3e3e5;--vp-c-bg-alt: #f6f6f7;--vp-c-border: rgba(60, 60, 67, .29);--vp-c-divider: rgba(60, 60, 67, .12);--vp-c-gutter: rgba(60, 60, 67, .12);--vp-c-neutral: var(--vp-c-black);--vp-c-neutral-inverse: var(--vp-c-white);--vp-c-text-1: var(--vp-c-text-light-1);--vp-c-text-2: var(--vp-c-text-light-2);--vp-c-text-3: var(--vp-c-text-light-3);--vp-c-text-inverse-1: var(--vp-c-text-dark-1);--vp-c-text-inverse-2: var(--vp-c-text-dark-2);--vp-c-text-inverse-3: var(--vp-c-text-dark-3);--vp-c-text-code: #476582;--vp-c-brand: var(--vp-c-green);--vp-c-brand-light: var(--vp-c-green-light);--vp-c-brand-lighter: var(--vp-c-green-lighter);--vp-c-brand-dark: var(--vp-c-green-dark);--vp-c-brand-darker: var(--vp-c-green-darker);--vp-c-mute: #f6f6f7;--vp-c-mute-light: #f9f9fc;--vp-c-mute-lighter: #ffffff;--vp-c-mute-dark: #e3e3e5;--vp-c-mute-darker: #d7d7d9}.dark{--vp-c-bg: #1e1e20;--vp-c-bg-elv: #252529;--vp-c-bg-elv-up: #313136;--vp-c-bg-elv-down: #1e1e20;--vp-c-bg-elv-mute: #313136;--vp-c-bg-soft: #252529;--vp-c-bg-soft-up: #313136;--vp-c-bg-soft-down: #1e1e20;--vp-c-bg-soft-mute: #313136;--vp-c-bg-alt: #161618;--vp-c-border: rgba(82, 82, 89, .68);--vp-c-divider: rgba(82, 82, 89, .32);--vp-c-gutter: #000000;--vp-c-neutral: var(--vp-c-white);--vp-c-neutral-inverse: var(--vp-c-black);--vp-c-text-1: var(--vp-c-text-dark-1);--vp-c-text-2: var(--vp-c-text-dark-2);--vp-c-text-3: var(--vp-c-text-dark-3);--vp-c-text-inverse-1: var(--vp-c-text-light-1);--vp-c-text-inverse-2: var(--vp-c-text-light-2);--vp-c-text-inverse-3: var(--vp-c-text-light-3);--vp-c-text-code: #c9def1;--vp-c-mute: #313136;--vp-c-mute-light: #3a3a3c;--vp-c-mute-lighter: #505053;--vp-c-mute-dark: #2c2c30;--vp-c-mute-darker: #252529}:root{--vp-font-family-base: "Chinese Quotes", "Inter var", "Inter", ui-sans-serif, system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Helvetica, Arial, "Noto Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";--vp-font-family-mono: ui-monospace, SFMono-Regular, "SF Mono", Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace}:root{--vp-shadow-1: 0 1px 2px rgba(0, 0, 0, .04), 0 1px 2px rgba(0, 0, 0, .06);--vp-shadow-2: 0 3px 12px rgba(0, 0, 0, .07), 0 1px 4px rgba(0, 0, 0, .07);--vp-shadow-3: 0 12px 32px rgba(0, 0, 0, .1), 0 2px 6px rgba(0, 0, 0, .08);--vp-shadow-4: 0 14px 44px rgba(0, 0, 0, .12), 0 3px 9px rgba(0, 0, 0, .12);--vp-shadow-5: 0 18px 56px rgba(0, 0, 0, .16), 0 4px 12px rgba(0, 0, 0, .16)}:root{--vp-z-index-local-nav: 10;--vp-z-index-nav: 20;--vp-z-index-layout-top: 30;--vp-z-index-backdrop: 40;--vp-z-index-sidebar: 50;--vp-z-index-footer: 60}:root{--vp-icon-copy: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' height='20' width='20' stroke='rgba(128,128,128,1)' stroke-width='2' viewBox='0 0 24 24'%3E%3Cpath stroke-linecap='round' stroke-linejoin='round' d='M9 5H7a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h10a2 2 0 0 0 2-2V7a2 2 0 0 0-2-2h-2M9 5a2 2 0 0 0 2 2h2a2 2 0 0 0 2-2M9 5a2 2 0 0 1 2-2h2a2 2 0 0 1 2 2'/%3E%3C/svg%3E");--vp-icon-copied: url("data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' fill='none' height='20' width='20' stroke='rgba(128,128,128,1)' stroke-width='2' viewBox='0 0 24 24'%3E%3Cpath stroke-linecap='round' stroke-linejoin='round' d='M9 5H7a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h10a2 2 0 0 0 2-2V7a2 2 0 0 0-2-2h-2M9 5a2 2 0 0 0 2 2h2a2 2 0 0 0 2-2M9 5a2 2 0 0 1 2-2h2a2 2 0 0 1 2 2m-6 9 2 2 4-4'/%3E%3C/svg%3E")}:root{--vp-layout-max-width: 1440px}:root{--vp-header-anchor-symbol: "#"}:root{--vp-code-line-height: 1.7;--vp-code-font-size: .875em;--vp-c-code-dimm: var(--vp-c-text-dark-3);--vp-code-block-color: var(--vp-c-text-dark-1);--vp-code-block-bg: #292b30;--vp-code-block-bg-light: #1e1e20;--vp-code-block-divider-color: #000000;--vp-code-line-highlight-color: rgba(0, 0, 0, .5);--vp-code-line-number-color: var(--vp-c-code-dimm);--vp-code-line-diff-add-color: var(--vp-c-green-dimm-2);--vp-code-line-diff-add-symbol-color: var(--vp-c-green);--vp-code-line-diff-remove-color: var(--vp-c-red-dimm-2);--vp-code-line-diff-remove-symbol-color: var(--vp-c-red);--vp-code-line-warning-color: var(--vp-c-yellow-dimm-2);--vp-code-line-error-color: var(--vp-c-red-dimm-2);--vp-code-copy-code-border-color: transparent;--vp-code-copy-code-bg: var(--vp-code-block-bg-light);--vp-code-copy-code-hover-border-color: var(--vp-c-divider);--vp-code-copy-code-hover-bg: var(--vp-code-block-bg-light);--vp-code-copy-code-active-text: var(--vp-c-text-dark-2);--vp-code-tab-divider: var(--vp-code-block-divider-color);--vp-code-tab-text-color: var(--vp-c-text-dark-2);--vp-code-tab-bg: var(--vp-code-block-bg);--vp-code-tab-hover-text-color: var(--vp-c-text-dark-1);--vp-code-tab-active-text-color: var(--vp-c-text-dark-1);--vp-code-tab-active-bar-color: var(--vp-c-brand)}.dark{--vp-code-block-bg: #161618}:root:not(.dark) .vp-adaptive-theme{--vp-c-code-dimm: var(--vp-c-text-2);--vp-code-block-color: var(--vp-c-text-1);--vp-code-block-bg: #f8f8f8;--vp-code-block-divider-color: var(--vp-c-divider);--vp-code-line-highlight-color: #ececec;--vp-code-line-number-color: var(--vp-c-code-dimm);--vp-code-copy-code-bg: #e2e2e2;--vp-code-copy-code-hover-bg: #dcdcdc;--vp-code-copy-code-active-text: var(--vp-c-text-2);--vp-code-tab-divider: var(--vp-c-divider);--vp-code-tab-text-color: var(--vp-c-text-2);--vp-code-tab-bg: var(--vp-code-block-bg);--vp-code-tab-hover-text-color: var(--vp-c-text-1);--vp-code-tab-active-text-color: var(--vp-c-text-1)}:root{--vp-button-brand-border: var(--vp-c-brand-lighter);--vp-button-brand-text: var(--vp-c-white);--vp-button-brand-bg: var(--vp-c-brand);--vp-button-brand-hover-border: var(--vp-c-brand-lighter);--vp-button-brand-hover-text: var(--vp-c-white);--vp-button-brand-hover-bg: var(--vp-c-brand-dark);--vp-button-brand-active-border: var(--vp-c-brand-lighter);--vp-button-brand-active-text: var(--vp-c-white);--vp-button-brand-active-bg: var(--vp-c-brand-darker);--vp-button-alt-border: var(--vp-c-border);--vp-button-alt-text: var(--vp-c-neutral);--vp-button-alt-bg: var(--vp-c-mute);--vp-button-alt-hover-border: var(--vp-c-border);--vp-button-alt-hover-text: var(--vp-c-neutral);--vp-button-alt-hover-bg: var(--vp-c-mute-dark);--vp-button-alt-active-border: var(--vp-c-border);--vp-button-alt-active-text: var(--vp-c-neutral);--vp-button-alt-active-bg: var(--vp-c-mute-darker);--vp-button-sponsor-border: var(--vp-c-gray-light-3);--vp-button-sponsor-text: var(--vp-c-text-light-2);--vp-button-sponsor-bg: transparent;--vp-button-sponsor-hover-border: var(--vp-c-sponsor);--vp-button-sponsor-hover-text: var(--vp-c-sponsor);--vp-button-sponsor-hover-bg: transparent;--vp-button-sponsor-active-border: var(--vp-c-sponsor);--vp-button-sponsor-active-text: var(--vp-c-sponsor);--vp-button-sponsor-active-bg: transparent}.dark{--vp-button-sponsor-border: var(--vp-c-gray-dark-1);--vp-button-sponsor-text: var(--vp-c-text-dark-2)}:root{--vp-custom-block-font-size: 14px;--vp-custom-block-code-font-size: 13px;--vp-custom-block-info-border: var(--vp-c-border);--vp-custom-block-info-text: var(--vp-c-text-2);--vp-custom-block-info-bg: var(--vp-c-bg-soft-up);--vp-custom-block-info-code-bg: var(--vp-c-bg-soft);--vp-custom-block-tip-border: var(--vp-c-green);--vp-custom-block-tip-text: var(--vp-c-green-dark);--vp-custom-block-tip-bg: var(--vp-c-bg-soft-up);--vp-custom-block-tip-code-bg: var(--vp-c-bg-soft);--vp-custom-block-warning-border: var(--vp-c-yellow);--vp-custom-block-warning-text: var(--vp-c-yellow);--vp-custom-block-warning-bg: var(--vp-c-bg-soft-up);--vp-custom-block-warning-code-bg: var(--vp-c-bg-soft);--vp-custom-block-danger-border: var(--vp-c-red);--vp-custom-block-danger-text: var(--vp-c-red);--vp-custom-block-danger-bg: var(--vp-c-bg-soft-up);--vp-custom-block-danger-code-bg: var(--vp-c-bg-soft);--vp-custom-block-details-border: var(--vp-custom-block-info-border);--vp-custom-block-details-text: var(--vp-custom-block-info-text);--vp-custom-block-details-bg: var(--vp-custom-block-info-bg);--vp-custom-block-details-code-bg: var(--vp-custom-block-details-bg)}:root{--vp-input-border-color: var(--vp-c-border);--vp-input-bg-color: var(--vp-c-bg-alt);--vp-input-hover-border-color: var(--vp-c-gray);--vp-input-switch-bg-color: var(--vp-c-mute)}:root{--vp-nav-height: 64px;--vp-nav-bg-color: var(--vp-c-bg);--vp-nav-screen-bg-color: var(--vp-c-bg)}:root{--vp-local-nav-bg-color: var(--vp-c-bg)}:root{--vp-sidebar-width: 272px;--vp-sidebar-bg-color: var(--vp-c-bg-alt)}:root{--vp-backdrop-bg-color: rgba(0, 0, 0, .6)}:root{--vp-home-hero-name-color: var(--vp-c-brand);--vp-home-hero-name-background: transparent;--vp-home-hero-image-background-image: none;--vp-home-hero-image-filter: none}:root{--vp-badge-info-border: var(--vp-c-border);--vp-badge-info-text: var(--vp-c-text-2);--vp-badge-info-bg: var(--vp-c-bg-soft-up);--vp-badge-tip-border: var(--vp-c-green-dark);--vp-badge-tip-text: var(--vp-c-green);--vp-badge-tip-bg: var(--vp-c-green-dimm-1);--vp-badge-warning-border: var(--vp-c-yellow-dark);--vp-badge-warning-text: var(--vp-c-yellow);--vp-badge-warning-bg: var(--vp-c-yellow-dimm-1);--vp-badge-danger-border: var(--vp-c-red-dark);--vp-badge-danger-text: var(--vp-c-red);--vp-badge-danger-bg: var(--vp-c-red-dimm-1)}:root{--vp-carbon-ads-text-color: var(--vp-c-text-1);--vp-carbon-ads-poweredby-color: var(--vp-c-text-2);--vp-carbon-ads-bg-color: var(--vp-c-bg-soft);--vp-carbon-ads-hover-text-color: var(--vp-c-brand);--vp-carbon-ads-hover-poweredby-color: var(--vp-c-text-1)}:root{--vp-local-search-bg: var(--vp-c-bg);--vp-local-search-result-bg: var(--vp-c-bg);--vp-local-search-result-border: var(--vp-c-divider);--vp-local-search-result-selected-bg: var(--vp-c-bg);--vp-local-search-result-selected-border: var(--vp-c-brand);--vp-local-search-highlight-bg: var(--vp-c-green-lighter);--vp-local-search-highlight-text: var(--vp-c-black)}*,:before,:after{box-sizing:border-box}html{line-height:1.4;font-size:16px;-webkit-text-size-adjust:100%}html.dark{color-scheme:dark}body{margin:0;width:100%;min-width:320px;min-height:100vh;line-height:24px;font-family:var(--vp-font-family-base);font-size:16px;font-weight:400;color:var(--vp-c-text-1);background-color:var(--vp-c-bg);direction:ltr;font-synthesis:style;text-rendering:optimizeLegibility;-webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale}h1,h2,h3,h4,h5,h6{margin:0;line-height:24px;font-size:16px;font-weight:400}p{margin:0}strong,b{font-weight:600}a,area,button,[role=button],input,label,select,summary,textarea{touch-action:manipulation}a{color:inherit;text-decoration:inherit}ol,ul{list-style:none;margin:0;padding:0}blockquote{margin:0}img,svg,video,canvas,audio,iframe,embed,object{display:block}figure{margin:0}img,video{max-width:100%;height:auto}button,input,optgroup,select,textarea{border:0;padding:0;line-height:inherit;color:inherit}button{padding:0;font-family:inherit;background-color:transparent;background-image:none}button:enabled,[role=button]:enabled{cursor:pointer}button:focus,button:focus-visible{outline:1px dotted;outline:4px auto -webkit-focus-ring-color}button:focus:not(:focus-visible){outline:none!important}input:focus,textarea:focus,select:focus{outline:none}table{border-collapse:collapse}input{background-color:transparent}input:-ms-input-placeholder,textarea:-ms-input-placeholder{color:var(--vp-c-text-3)}input::-ms-input-placeholder,textarea::-ms-input-placeholder{color:var(--vp-c-text-3)}input::placeholder,textarea::placeholder{color:var(--vp-c-text-3)}input::-webkit-outer-spin-button,input::-webkit-inner-spin-button{-webkit-appearance:none;margin:0}input[type=number]{-moz-appearance:textfield}textarea{resize:vertical}select{-webkit-appearance:none}fieldset{margin:0;padding:0}h1,h2,h3,h4,h5,h6,li,p{overflow-wrap:break-word}vite-error-overlay{z-index:9999}.visually-hidden{position:absolute;width:1px;height:1px;white-space:nowrap;clip:rect(0 0 0 0);clip-path:inset(50%);overflow:hidden}.custom-block{border:1px solid transparent;border-radius:8px;padding:16px 16px 8px;line-height:24px;font-size:var(--vp-custom-block-font-size);color:var(--vp-c-text-2)}.custom-block.info{border-color:var(--vp-custom-block-info-border);color:var(--vp-custom-block-info-text);background-color:var(--vp-custom-block-info-bg)}.custom-block.custom-block th,.custom-block.custom-block blockquote>p{font-size:var(--vp-custom-block-font-size);color:inherit}.custom-block.info code{background-color:var(--vp-custom-block-info-code-bg)}.custom-block.tip{border-color:var(--vp-custom-block-tip-border);color:var(--vp-custom-block-tip-text);background-color:var(--vp-custom-block-tip-bg)}.custom-block.tip code{background-color:var(--vp-custom-block-tip-code-bg)}.custom-block.warning{border-color:var(--vp-custom-block-warning-border);color:var(--vp-custom-block-warning-text);background-color:var(--vp-custom-block-warning-bg)}.custom-block.warning code{background-color:var(--vp-custom-block-warning-code-bg)}.custom-block.danger{border-color:var(--vp-custom-block-danger-border);color:var(--vp-custom-block-danger-text);background-color:var(--vp-custom-block-danger-bg)}.custom-block.danger code{background-color:var(--vp-custom-block-danger-code-bg)}.custom-block.details{border-color:var(--vp-custom-block-details-border);color:var(--vp-custom-block-details-text);background-color:var(--vp-custom-block-details-bg)}.custom-block.details code{background-color:var(--vp-custom-block-details-code-bg)}.custom-block-title{font-weight:600}.custom-block p+p{margin:8px 0}.custom-block.details summary{margin:0 0 8px;font-weight:700;cursor:pointer}.custom-block.details summary+p{margin:8px 0}.custom-block a{color:inherit;font-weight:600}.custom-block a:hover{text-decoration:underline}.custom-block code{font-size:var(--vp-custom-block-code-font-size)}.dark .vp-code-light{display:none}html:not(.dark) .vp-code-dark{display:none}.vp-code-group{margin-top:16px}.vp-code-group .tabs{position:relative;display:flex;margin-right:-24px;margin-left:-24px;padding:0 12px;background-color:var(--vp-code-tab-bg);overflow-x:auto;overflow-y:hidden}.vp-code-group .tabs:after{position:absolute;right:0;bottom:0;left:0;height:1px;background-color:var(--vp-code-tab-divider);content:""}@media (min-width: 640px){.vp-code-group .tabs{margin-right:0;margin-left:0;border-radius:8px 8px 0 0}}.vp-code-group .tabs input{position:absolute;opacity:0;pointer-events:none}.vp-code-group .tabs label{position:relative;display:inline-block;border-bottom:1px solid transparent;padding:0 12px;line-height:48px;font-size:14px;font-weight:500;color:var(--vp-code-tab-text-color);white-space:nowrap;cursor:pointer;transition:color .25s}.vp-code-group .tabs label:after{position:absolute;right:8px;bottom:-1px;left:8px;z-index:10;height:1px;content:"";background-color:transparent;transition:background-color .25s}.vp-code-group label:hover{color:var(--vp-code-tab-hover-text-color)}.vp-code-group input:checked+label{color:var(--vp-code-tab-active-text-color)}.vp-code-group input:checked+label:after{background-color:var(--vp-code-tab-active-bar-color)}.vp-code-group div[class*=language-]{display:none;margin-top:0!important;border-top-left-radius:0!important;border-top-right-radius:0!important}.vp-code-group div[class*=language-].active{display:block}.vp-doc h1,.vp-doc h2,.vp-doc h3,.vp-doc h4,.vp-doc h5,.vp-doc h6{position:relative;font-weight:600;outline:none}.vp-doc h1{letter-spacing:-.02em;line-height:40px;font-size:28px}.vp-doc h2{margin:48px 0 16px;border-top:1px solid var(--vp-c-divider);padding-top:24px;letter-spacing:-.02em;line-height:32px;font-size:24px}.vp-doc h3{margin:32px 0 0;letter-spacing:-.01em;line-height:28px;font-size:20px}.vp-doc .header-anchor{float:left;margin-left:-.87em;padding-right:.23em;font-weight:500;user-select:none;opacity:0;transition:color .25s,opacity .25s}.vp-doc .header-anchor:before{content:var(--vp-header-anchor-symbol)}.vp-doc h1:hover .header-anchor,.vp-doc h1 .header-anchor:focus,.vp-doc h2:hover .header-anchor,.vp-doc h2 .header-anchor:focus,.vp-doc h3:hover .header-anchor,.vp-doc h3 .header-anchor:focus,.vp-doc h4:hover .header-anchor,.vp-doc h4 .header-anchor:focus,.vp-doc h5:hover .header-anchor,.vp-doc h5 .header-anchor:focus,.vp-doc h6:hover .header-anchor,.vp-doc h6 .header-anchor:focus{opacity:1}@media (min-width: 768px){.vp-doc h1{letter-spacing:-.02em;line-height:40px;font-size:32px}}.vp-doc p,.vp-doc summary{margin:16px 0}.vp-doc p{line-height:28px}.vp-doc blockquote{margin:16px 0;border-left:2px solid var(--vp-c-divider);padding-left:16px;transition:border-color .5s}.vp-doc blockquote>p{margin:0;font-size:16px;color:var(--vp-c-text-2);transition:color .5s}.vp-doc a{font-weight:500;color:var(--vp-c-brand);text-decoration-style:dotted;transition:color .25s}.vp-doc a:hover{text-decoration:underline}.vp-doc strong{font-weight:600}.vp-doc ul,.vp-doc ol{padding-left:1.25rem;margin:16px 0}.vp-doc ul{list-style:disc}.vp-doc ol{list-style:decimal}.vp-doc li+li{margin-top:8px}.vp-doc li>ol,.vp-doc li>ul{margin:8px 0 0}.vp-doc table{display:block;border-collapse:collapse;margin:20px 0;overflow-x:auto}.vp-doc tr{border-top:1px solid var(--vp-c-divider);transition:background-color .5s}.vp-doc tr:nth-child(2n){background-color:var(--vp-c-bg-soft)}.vp-doc th,.vp-doc td{border:1px solid var(--vp-c-divider);padding:8px 16px}.vp-doc th{text-align:left;font-size:14px;font-weight:600;color:var(--vp-c-text-2);background-color:var(--vp-c-bg-soft)}.vp-doc td{font-size:14px}.vp-doc hr{margin:16px 0;border:none;border-top:1px solid var(--vp-c-divider)}.vp-doc .custom-block{margin:16px 0}.vp-doc .custom-block p{margin:8px 0;line-height:24px}.vp-doc .custom-block p:first-child{margin:0}.vp-doc .custom-block a{color:inherit;font-weight:600}.vp-doc .custom-block a:hover{text-decoration:underline}.vp-doc .custom-block code{font-size:var(--vp-custom-block-code-font-size);font-weight:700;color:inherit}.vp-doc .custom-block div[class*=language-]{margin:8px 0}.vp-doc .custom-block div[class*=language-] code{font-weight:400;background-color:transparent}.vp-doc :not(pre,h1,h2,h3,h4,h5,h6)>code{font-size:var(--vp-code-font-size)}.vp-doc :not(pre)>code{border-radius:4px;padding:3px 6px;color:var(--vp-c-text-code);background-color:var(--vp-c-mute);transition:color .5s,background-color .5s}.vp-doc h1>code,.vp-doc h2>code,.vp-doc h3>code{font-size:.9em}.vp-doc a>code{color:var(--vp-c-brand);transition:color .25s}.vp-doc a:hover>code{color:var(--vp-c-brand-dark)}.vp-doc div[class*=language-]{position:relative;margin:16px -24px;background-color:var(--vp-code-block-bg);overflow-x:auto;transition:background-color .5s}@media (min-width: 640px){.vp-doc div[class*=language-]{border-radius:8px;margin:16px 0}}@media (max-width: 639px){.vp-doc li div[class*=language-]{border-radius:8px 0 0 8px}}.vp-doc div[class*=language-]+div[class*=language-],.vp-doc div[class$=-api]+div[class*=language-],.vp-doc div[class*=language-]+div[class$=-api]>div[class*=language-]{margin-top:-8px}.vp-doc [class*=language-] pre,.vp-doc [class*=language-] code{direction:ltr;text-align:left;white-space:pre;word-spacing:normal;word-break:normal;word-wrap:normal;-moz-tab-size:4;-o-tab-size:4;tab-size:4;-webkit-hyphens:none;-moz-hyphens:none;-ms-hyphens:none;hyphens:none}.vp-doc [class*=language-] pre{position:relative;z-index:1;margin:0;padding:20px 0;background:transparent;overflow-x:auto}.vp-doc [class*=language-] code{display:block;padding:0 24px;width:fit-content;min-width:100%;line-height:var(--vp-code-line-height);font-size:var(--vp-code-font-size);color:var(--vp-code-block-color);transition:color .5s}.vp-doc [class*=language-] code .highlighted{background-color:var(--vp-code-line-highlight-color);transition:background-color .5s;margin:0 -24px;padding:0 24px;width:calc(100% + 48px);display:inline-block}.vp-doc [class*=language-] code .highlighted.error{background-color:var(--vp-code-line-error-color)}.vp-doc [class*=language-] code .highlighted.warning{background-color:var(--vp-code-line-warning-color)}.vp-doc [class*=language-] code .diff{transition:background-color .5s;margin:0 -24px;padding:0 24px;width:calc(100% + 48px);display:inline-block}.vp-doc [class*=language-] code .diff:before{position:absolute;left:10px}.vp-doc [class*=language-] .has-focused-lines .line:not(.has-focus){filter:blur(.095rem);opacity:.4;transition:filter .35s,opacity .35s}.vp-doc [class*=language-] .has-focused-lines .line:not(.has-focus){opacity:.7;transition:filter .35s,opacity .35s}.vp-doc [class*=language-]:hover .has-focused-lines .line:not(.has-focus){filter:blur(0);opacity:1}.vp-doc [class*=language-] code .diff.remove{background-color:var(--vp-code-line-diff-remove-color);opacity:.7}.vp-doc [class*=language-] code .diff.remove:before{content:"-";color:var(--vp-code-line-diff-remove-symbol-color)}.vp-doc [class*=language-] code .diff.add{background-color:var(--vp-code-line-diff-add-color)}.vp-doc [class*=language-] code .diff.add:before{content:"+";color:var(--vp-code-line-diff-add-symbol-color)}.vp-doc div[class*=language-].line-numbers-mode{padding-left:32px}.vp-doc .line-numbers-wrapper{position:absolute;top:0;bottom:0;left:0;z-index:3;border-right:1px solid var(--vp-code-block-divider-color);padding-top:20px;width:32px;text-align:center;font-family:var(--vp-font-family-mono);line-height:var(--vp-code-line-height);font-size:var(--vp-code-font-size);color:var(--vp-code-line-number-color);transition:border-color .5s,color .5s}.vp-doc [class*=language-]>button.copy{direction:ltr;position:absolute;top:12px;right:12px;z-index:3;border:1px solid var(--vp-code-copy-code-border-color);border-radius:4px;width:40px;height:40px;background-color:var(--vp-code-copy-code-bg);opacity:0;cursor:pointer;background-image:var(--vp-icon-copy);background-position:50%;background-size:20px;background-repeat:no-repeat;transition:border-color .25s,background-color .25s,opacity .25s}.vp-doc [class*=language-]:hover>button.copy,.vp-doc [class*=language-]>button.copy:focus{opacity:1}.vp-doc [class*=language-]>button.copy:hover,.vp-doc [class*=language-]>button.copy.copied{border-color:var(--vp-code-copy-code-hover-border-color);background-color:var(--vp-code-copy-code-hover-bg)}.vp-doc [class*=language-]>button.copy.copied,.vp-doc [class*=language-]>button.copy:hover.copied{border-radius:0 4px 4px 0;background-color:var(--vp-code-copy-code-hover-bg);background-image:var(--vp-icon-copied)}.vp-doc [class*=language-]>button.copy.copied:before,.vp-doc [class*=language-]>button.copy:hover.copied:before{position:relative;top:-1px;left:-65px;display:flex;justify-content:center;align-items:center;border:1px solid var(--vp-code-copy-code-hover-border-color);border-right:0;border-radius:4px 0 0 4px;width:64px;height:40px;text-align:center;font-size:12px;font-weight:500;color:var(--vp-code-copy-code-active-text);background-color:var(--vp-code-copy-code-hover-bg);white-space:nowrap;content:"Copied"}.vp-doc [class*=language-]>span.lang{position:absolute;top:2px;right:8px;z-index:2;font-size:12px;font-weight:500;color:var(--vp-c-code-dimm);transition:color .4s,opacity .4s}.vp-doc [class*=language-]:hover>button.copy+span.lang,.vp-doc [class*=language-]>button.copy:focus+span.lang{opacity:0}.vp-doc .VPTeamMembers{margin-top:24px}.vp-doc .VPTeamMembers.small.count-1 .container{margin:0!important;max-width:calc((100% - 24px)/2)!important}.vp-doc .VPTeamMembers.small.count-2 .container,.vp-doc .VPTeamMembers.small.count-3 .container{max-width:100%!important}.vp-doc .VPTeamMembers.medium.count-1 .container{margin:0!important;max-width:calc((100% - 24px)/2)!important}.vp-sponsor{border-radius:16px;overflow:hidden}.vp-sponsor.aside{border-radius:12px}.vp-sponsor-section+.vp-sponsor-section{margin-top:4px}.vp-sponsor-tier{margin-bottom:4px;text-align:center;letter-spacing:1px;line-height:24px;width:100%;font-weight:600;color:var(--vp-c-text-2);background-color:var(--vp-c-bg-soft)}.vp-sponsor.normal .vp-sponsor-tier{padding:13px 0 11px;font-size:14px}.vp-sponsor.aside .vp-sponsor-tier{padding:9px 0 7px;font-size:12px}.vp-sponsor-grid+.vp-sponsor-tier{margin-top:4px}.vp-sponsor-grid{display:flex;flex-wrap:wrap;gap:4px}.vp-sponsor-grid.xmini .vp-sponsor-grid-link{height:64px}.vp-sponsor-grid.xmini .vp-sponsor-grid-image{max-width:64px;max-height:22px}.vp-sponsor-grid.mini .vp-sponsor-grid-link{height:72px}.vp-sponsor-grid.mini .vp-sponsor-grid-image{max-width:96px;max-height:24px}.vp-sponsor-grid.small .vp-sponsor-grid-link{height:96px}.vp-sponsor-grid.small .vp-sponsor-grid-image{max-width:96px;max-height:24px}.vp-sponsor-grid.medium .vp-sponsor-grid-link{height:112px}.vp-sponsor-grid.medium .vp-sponsor-grid-image{max-width:120px;max-height:36px}.vp-sponsor-grid.big .vp-sponsor-grid-link{height:184px}.vp-sponsor-grid.big .vp-sponsor-grid-image{max-width:192px;max-height:56px}.vp-sponsor-grid[data-vp-grid="2"] .vp-sponsor-grid-item{width:calc((100% - 4px)/2)}.vp-sponsor-grid[data-vp-grid="3"] .vp-sponsor-grid-item{width:calc((100% - 4px * 2) / 3)}.vp-sponsor-grid[data-vp-grid="4"] .vp-sponsor-grid-item{width:calc((100% - 12px)/4)}.vp-sponsor-grid[data-vp-grid="5"] .vp-sponsor-grid-item{width:calc((100% - 16px)/5)}.vp-sponsor-grid[data-vp-grid="6"] .vp-sponsor-grid-item{width:calc((100% - 4px * 5) / 6)}.vp-sponsor-grid-item{flex-shrink:0;width:100%;background-color:var(--vp-c-bg-soft);transition:background-color .25s}.vp-sponsor-grid-item:hover{background-color:var(--vp-c-bg-soft-down)}.vp-sponsor-grid-item:hover .vp-sponsor-grid-image{filter:grayscale(0) invert(0)}.vp-sponsor-grid-item.empty:hover{background-color:var(--vp-c-bg-soft)}.dark .vp-sponsor-grid-item:hover{background-color:var(--vp-c-white)}.dark .vp-sponsor-grid-item.empty:hover{background-color:var(--vp-c-bg-soft)}.vp-sponsor-grid-link{display:flex}.vp-sponsor-grid-box{display:flex;justify-content:center;align-items:center;width:100%}.vp-sponsor-grid-image{max-width:100%;filter:grayscale(1);transition:filter .25s}.dark .vp-sponsor-grid-image{filter:grayscale(1) invert(1)}.VPBadge[data-v-b8591dbf]{display:inline-block;margin-left:2px;border:1px solid transparent;border-radius:10px;padding:0 8px;line-height:18px;font-size:12px;font-weight:600;transform:translateY(-2px)}h1 .VPBadge[data-v-b8591dbf],h2 .VPBadge[data-v-b8591dbf],h3 .VPBadge[data-v-b8591dbf],h4 .VPBadge[data-v-b8591dbf],h5 .VPBadge[data-v-b8591dbf],h6 .VPBadge[data-v-b8591dbf]{vertical-align:top}h2 .VPBadge[data-v-b8591dbf]{border-radius:11px;line-height:20px}.VPBadge.info[data-v-b8591dbf]{border-color:var(--vp-badge-info-border);color:var(--vp-badge-info-text);background-color:var(--vp-badge-info-bg)}.VPBadge.tip[data-v-b8591dbf]{border-color:var(--vp-badge-tip-border);color:var(--vp-badge-tip-text);background-color:var(--vp-badge-tip-bg)}.VPBadge.warning[data-v-b8591dbf]{border-color:var(--vp-badge-warning-border);color:var(--vp-badge-warning-text);background-color:var(--vp-badge-warning-bg)}.VPBadge.danger[data-v-b8591dbf]{border-color:var(--vp-badge-danger-border);color:var(--vp-badge-danger-text);background-color:var(--vp-badge-danger-bg)}.VPSkipLink[data-v-aa63c34f]{top:8px;left:8px;padding:8px 16px;z-index:999;border-radius:8px;font-size:12px;font-weight:700;text-decoration:none;color:var(--vp-c-brand);box-shadow:var(--vp-shadow-3);background-color:var(--vp-c-bg)}.VPSkipLink[data-v-aa63c34f]:focus{height:auto;width:auto;clip:auto;clip-path:none}.dark .VPSkipLink[data-v-aa63c34f]{color:var(--vp-c-green)}@media (min-width: 1280px){.VPSkipLink[data-v-aa63c34f]{top:14px;left:16px}}.VPBackdrop[data-v-d0d4b99a]{position:fixed;top:0;right:0;bottom:0;left:0;z-index:var(--vp-z-index-backdrop);background:var(--vp-backdrop-bg-color);transition:opacity .5s}.VPBackdrop.fade-enter-from[data-v-d0d4b99a],.VPBackdrop.fade-leave-to[data-v-d0d4b99a]{opacity:0}.VPBackdrop.fade-leave-active[data-v-d0d4b99a]{transition-duration:.25s}@media (min-width: 1280px){.VPBackdrop[data-v-d0d4b99a]{display:none}}html:not(.dark) .VPImage.dark[data-v-6f75ecd0]{display:none}.dark .VPImage.light[data-v-6f75ecd0]{display:none}.title[data-v-75402b34]{display:flex;align-items:center;border-bottom:1px solid transparent;width:100%;height:var(--vp-nav-height);font-size:16px;font-weight:600;color:var(--vp-c-text-1);transition:opacity .25s}.title[data-v-75402b34]:hover{opacity:.6}@media (min-width: 960px){.title[data-v-75402b34]{flex-shrink:0}.VPNavBarTitle.has-sidebar .title[data-v-75402b34]{border-bottom-color:var(--vp-c-divider)}}[data-v-75402b34] .logo{margin-right:8px;height:24px}/*! @docsearch/css 3.3.4 | MIT License | © Algolia, Inc. and contributors | https://docsearch.algolia.com */:root{--docsearch-primary-color:#5468ff;--docsearch-text-color:#1c1e21;--docsearch-spacing:12px;--docsearch-icon-stroke-width:1.4;--docsearch-highlight-color:var(--docsearch-primary-color);--docsearch-muted-color:#969faf;--docsearch-container-background:rgba(101,108,133,.8);--docsearch-logo-color:#5468ff;--docsearch-modal-width:560px;--docsearch-modal-height:600px;--docsearch-modal-background:#f5f6f7;--docsearch-modal-shadow:inset 1px 1px 0 0 hsla(0,0%,100%,.5),0 3px 8px 0 #555a64;--docsearch-searchbox-height:56px;--docsearch-searchbox-background:#ebedf0;--docsearch-searchbox-focus-background:#fff;--docsearch-searchbox-shadow:inset 0 0 0 2px var(--docsearch-primary-color);--docsearch-hit-height:56px;--docsearch-hit-color:#444950;--docsearch-hit-active-color:#fff;--docsearch-hit-background:#fff;--docsearch-hit-shadow:0 1px 3px 0 #d4d9e1;--docsearch-key-gradient:linear-gradient(-225deg,#d5dbe4,#f8f8f8);--docsearch-key-shadow:inset 0 -2px 0 0 #cdcde6,inset 0 0 1px 1px #fff,0 1px 2px 1px rgba(30,35,90,.4);--docsearch-footer-height:44px;--docsearch-footer-background:#fff;--docsearch-footer-shadow:0 -1px 0 0 #e0e3e8,0 -3px 6px 0 rgba(69,98,155,.12)}html[data-theme=dark]{--docsearch-text-color:#f5f6f7;--docsearch-container-background:rgba(9,10,17,.8);--docsearch-modal-background:#15172a;--docsearch-modal-shadow:inset 1px 1px 0 0 #2c2e40,0 3px 8px 0 #000309;--docsearch-searchbox-background:#090a11;--docsearch-searchbox-focus-background:#000;--docsearch-hit-color:#bec3c9;--docsearch-hit-shadow:none;--docsearch-hit-background:#090a11;--docsearch-key-gradient:linear-gradient(-26.5deg,#565872,#31355b);--docsearch-key-shadow:inset 0 -2px 0 0 #282d55,inset 0 0 1px 1px #51577d,0 2px 2px 0 rgba(3,4,9,.3);--docsearch-footer-background:#1e2136;--docsearch-footer-shadow:inset 0 1px 0 0 rgba(73,76,106,.5),0 -4px 8px 0 rgba(0,0,0,.2);--docsearch-logo-color:#fff;--docsearch-muted-color:#7f8497}.DocSearch-Button{align-items:center;background:var(--docsearch-searchbox-background);border:0;border-radius:40px;color:var(--docsearch-muted-color);cursor:pointer;display:flex;font-weight:500;height:36px;justify-content:space-between;margin:0 0 0 16px;padding:0 8px;user-select:none}.DocSearch-Button:active,.DocSearch-Button:focus,.DocSearch-Button:hover{background:var(--docsearch-searchbox-focus-background);box-shadow:var(--docsearch-searchbox-shadow);color:var(--docsearch-text-color);outline:none}.DocSearch-Button-Container{align-items:center;display:flex}.DocSearch-Search-Icon{stroke-width:1.6}.DocSearch-Button .DocSearch-Search-Icon{color:var(--docsearch-text-color)}.DocSearch-Button-Placeholder{font-size:1rem;padding:0 12px 0 6px}.DocSearch-Button-Keys{display:flex;min-width:calc(40px + .8em)}.DocSearch-Button-Key{align-items:center;background:var(--docsearch-key-gradient);border-radius:3px;box-shadow:var(--docsearch-key-shadow);color:var(--docsearch-muted-color);display:flex;height:18px;justify-content:center;margin-right:.4em;position:relative;padding:0 0 2px;border:0;top:-1px;width:20px}@media (max-width:768px){.DocSearch-Button-Keys,.DocSearch-Button-Placeholder{display:none}}.DocSearch--active{overflow:hidden!important}.DocSearch-Container,.DocSearch-Container *{box-sizing:border-box}.DocSearch-Container{background-color:var(--docsearch-container-background);height:100vh;left:0;position:fixed;top:0;width:100vw;z-index:200}.DocSearch-Container a{text-decoration:none}.DocSearch-Link{appearance:none;background:none;border:0;color:var(--docsearch-highlight-color);cursor:pointer;font:inherit;margin:0;padding:0}.DocSearch-Modal{background:var(--docsearch-modal-background);border-radius:6px;box-shadow:var(--docsearch-modal-shadow);flex-direction:column;margin:60px auto auto;max-width:var(--docsearch-modal-width);position:relative}.DocSearch-SearchBar{display:flex;padding:var(--docsearch-spacing) var(--docsearch-spacing) 0}.DocSearch-Form{align-items:center;background:var(--docsearch-searchbox-focus-background);border-radius:4px;box-shadow:var(--docsearch-searchbox-shadow);display:flex;height:var(--docsearch-searchbox-height);margin:0;padding:0 var(--docsearch-spacing);position:relative;width:100%}.DocSearch-Input{appearance:none;background:transparent;border:0;color:var(--docsearch-text-color);flex:1;font:inherit;font-size:1.2em;height:100%;outline:none;padding:0 0 0 8px;width:80%}.DocSearch-Input::placeholder{color:var(--docsearch-muted-color);opacity:1}.DocSearch-Input::-webkit-search-cancel-button,.DocSearch-Input::-webkit-search-decoration,.DocSearch-Input::-webkit-search-results-button,.DocSearch-Input::-webkit-search-results-decoration{display:none}.DocSearch-LoadingIndicator,.DocSearch-MagnifierLabel,.DocSearch-Reset{margin:0;padding:0}.DocSearch-MagnifierLabel,.DocSearch-Reset{align-items:center;color:var(--docsearch-highlight-color);display:flex;justify-content:center}.DocSearch-Container--Stalled .DocSearch-MagnifierLabel,.DocSearch-LoadingIndicator{display:none}.DocSearch-Container--Stalled .DocSearch-LoadingIndicator{align-items:center;color:var(--docsearch-highlight-color);display:flex;justify-content:center}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Reset{animation:none;appearance:none;background:none;border:0;border-radius:50%;color:var(--docsearch-icon-color);cursor:pointer;right:0;stroke-width:var(--docsearch-icon-stroke-width)}}.DocSearch-Reset{animation:fade-in .1s ease-in forwards;appearance:none;background:none;border:0;border-radius:50%;color:var(--docsearch-icon-color);cursor:pointer;padding:2px;right:0;stroke-width:var(--docsearch-icon-stroke-width)}.DocSearch-Reset[hidden]{display:none}.DocSearch-Reset:hover{color:var(--docsearch-highlight-color)}.DocSearch-LoadingIndicator svg,.DocSearch-MagnifierLabel svg{height:24px;width:24px}.DocSearch-Cancel{display:none}.DocSearch-Dropdown{max-height:calc(var(--docsearch-modal-height) - var(--docsearch-searchbox-height) - var(--docsearch-spacing) - var(--docsearch-footer-height));min-height:var(--docsearch-spacing);overflow-y:auto;overflow-y:overlay;padding:0 var(--docsearch-spacing);scrollbar-color:var(--docsearch-muted-color) var(--docsearch-modal-background);scrollbar-width:thin}.DocSearch-Dropdown::-webkit-scrollbar{width:12px}.DocSearch-Dropdown::-webkit-scrollbar-track{background:transparent}.DocSearch-Dropdown::-webkit-scrollbar-thumb{background-color:var(--docsearch-muted-color);border:3px solid var(--docsearch-modal-background);border-radius:20px}.DocSearch-Dropdown ul{list-style:none;margin:0;padding:0}.DocSearch-Label{font-size:.75em;line-height:1.6em}.DocSearch-Help,.DocSearch-Label{color:var(--docsearch-muted-color)}.DocSearch-Help{font-size:.9em;margin:0;user-select:none}.DocSearch-Title{font-size:1.2em}.DocSearch-Logo a{display:flex}.DocSearch-Logo svg{color:var(--docsearch-logo-color);margin-left:8px}.DocSearch-Hits:last-of-type{margin-bottom:24px}.DocSearch-Hits mark{background:none;color:var(--docsearch-highlight-color)}.DocSearch-HitsFooter{color:var(--docsearch-muted-color);display:flex;font-size:.85em;justify-content:center;margin-bottom:var(--docsearch-spacing);padding:var(--docsearch-spacing)}.DocSearch-HitsFooter a{border-bottom:1px solid;color:inherit}.DocSearch-Hit{border-radius:4px;display:flex;padding-bottom:4px;position:relative}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit--deleting{transition:none}}.DocSearch-Hit--deleting{opacity:0;transition:all .25s linear}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit--favoriting{transition:none}}.DocSearch-Hit--favoriting{transform:scale(0);transform-origin:top center;transition:all .25s linear;transition-delay:.25s}.DocSearch-Hit a{background:var(--docsearch-hit-background);border-radius:4px;box-shadow:var(--docsearch-hit-shadow);display:block;padding-left:var(--docsearch-spacing);width:100%}.DocSearch-Hit-source{background:var(--docsearch-modal-background);color:var(--docsearch-highlight-color);font-size:.85em;font-weight:600;line-height:32px;margin:0 -4px;padding:8px 4px 0;position:sticky;top:0;z-index:10}.DocSearch-Hit-Tree{color:var(--docsearch-muted-color);height:var(--docsearch-hit-height);opacity:.5;stroke-width:var(--docsearch-icon-stroke-width);width:24px}.DocSearch-Hit[aria-selected=true] a{background-color:var(--docsearch-highlight-color)}.DocSearch-Hit[aria-selected=true] mark{text-decoration:underline}.DocSearch-Hit-Container{align-items:center;color:var(--docsearch-hit-color);display:flex;flex-direction:row;height:var(--docsearch-hit-height);padding:0 var(--docsearch-spacing) 0 0}.DocSearch-Hit-icon{height:20px;width:20px}.DocSearch-Hit-action,.DocSearch-Hit-icon{color:var(--docsearch-muted-color);stroke-width:var(--docsearch-icon-stroke-width)}.DocSearch-Hit-action{align-items:center;display:flex;height:22px;width:22px}.DocSearch-Hit-action svg{display:block;height:18px;width:18px}.DocSearch-Hit-action+.DocSearch-Hit-action{margin-left:6px}.DocSearch-Hit-action-button{appearance:none;background:none;border:0;border-radius:50%;color:inherit;cursor:pointer;padding:2px}svg.DocSearch-Hit-Select-Icon{display:none}.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-Select-Icon{display:block}.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{background:rgba(0,0,0,.2);transition:background-color .1s ease-in}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{transition:none}}.DocSearch-Hit-action-button:focus path,.DocSearch-Hit-action-button:hover path{fill:#fff}.DocSearch-Hit-content-wrapper{display:flex;flex:1 1 auto;flex-direction:column;font-weight:500;justify-content:center;line-height:1.2em;margin:0 8px;overflow-x:hidden;position:relative;text-overflow:ellipsis;white-space:nowrap;width:80%}.DocSearch-Hit-title{font-size:.9em}.DocSearch-Hit-path{color:var(--docsearch-muted-color);font-size:.75em}.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-action,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-icon,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-path,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-text,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-title,.DocSearch-Hit[aria-selected=true] .DocSearch-Hit-Tree,.DocSearch-Hit[aria-selected=true] mark{color:var(--docsearch-hit-active-color)!important}@media screen and (prefers-reduced-motion:reduce){.DocSearch-Hit-action-button:focus,.DocSearch-Hit-action-button:hover{background:rgba(0,0,0,.2);transition:none}}.DocSearch-ErrorScreen,.DocSearch-NoResults,.DocSearch-StartScreen{font-size:.9em;margin:0 auto;padding:36px 0;text-align:center;width:80%}.DocSearch-Screen-Icon{color:var(--docsearch-muted-color);padding-bottom:12px}.DocSearch-NoResults-Prefill-List{display:inline-block;padding-bottom:24px;text-align:left}.DocSearch-NoResults-Prefill-List ul{display:inline-block;padding:8px 0 0}.DocSearch-NoResults-Prefill-List li{list-style-position:inside;list-style-type:"» "}.DocSearch-Prefill{appearance:none;background:none;border:0;border-radius:1em;color:var(--docsearch-highlight-color);cursor:pointer;display:inline-block;font-size:1em;font-weight:700;padding:0}.DocSearch-Prefill:focus,.DocSearch-Prefill:hover{outline:none;text-decoration:underline}.DocSearch-Footer{align-items:center;background:var(--docsearch-footer-background);border-radius:0 0 8px 8px;box-shadow:var(--docsearch-footer-shadow);display:flex;flex-direction:row-reverse;flex-shrink:0;height:var(--docsearch-footer-height);justify-content:space-between;padding:0 var(--docsearch-spacing);position:relative;user-select:none;width:100%;z-index:300}.DocSearch-Commands{color:var(--docsearch-muted-color);display:flex;list-style:none;margin:0;padding:0}.DocSearch-Commands li{align-items:center;display:flex}.DocSearch-Commands li:not(:last-of-type){margin-right:.8em}.DocSearch-Commands-Key{align-items:center;background:var(--docsearch-key-gradient);border-radius:2px;box-shadow:var(--docsearch-key-shadow);display:flex;height:18px;justify-content:center;margin-right:.4em;padding:0 0 1px;color:var(--docsearch-muted-color);border:0;width:20px}@media (max-width:768px){:root{--docsearch-spacing:10px;--docsearch-footer-height:40px}.DocSearch-Dropdown{height:100%}.DocSearch-Container{height:100vh;height:-webkit-fill-available;height:calc(var(--docsearch-vh, 1vh)*100);position:absolute}.DocSearch-Footer{border-radius:0;bottom:0;position:absolute}.DocSearch-Hit-content-wrapper{display:flex;position:relative;width:80%}.DocSearch-Modal{border-radius:0;box-shadow:none;height:100vh;height:-webkit-fill-available;height:calc(var(--docsearch-vh, 1vh)*100);margin:0;max-width:100%;width:100%}.DocSearch-Dropdown{max-height:calc(var(--docsearch-vh, 1vh)*100 - var(--docsearch-searchbox-height) - var(--docsearch-spacing) - var(--docsearch-footer-height))}.DocSearch-Cancel{appearance:none;background:none;border:0;color:var(--docsearch-highlight-color);cursor:pointer;display:inline-block;flex:none;font:inherit;font-size:1em;font-weight:500;margin-left:var(--docsearch-spacing);outline:none;overflow:hidden;padding:0;user-select:none;white-space:nowrap}.DocSearch-Commands,.DocSearch-Hit-Tree{display:none}}@keyframes fade-in{0%{opacity:0}to{opacity:1}}.DocSearch{--docsearch-primary-color: var(--vp-c-brand);--docsearch-highlight-color: var(--docsearch-primary-color);--docsearch-text-color: var(--vp-c-text-1);--docsearch-muted-color: var(--vp-c-text-2);--docsearch-searchbox-shadow: none;--docsearch-searchbox-focus-background: transparent;--docsearch-key-gradient: transparent;--docsearch-key-shadow: none;--docsearch-modal-background: var(--vp-c-bg-soft);--docsearch-footer-background: var(--vp-c-bg)}.dark .DocSearch{--docsearch-modal-shadow: none;--docsearch-footer-shadow: none;--docsearch-logo-color: var(--vp-c-text-2);--docsearch-hit-background: var(--vp-c-bg-soft-mute);--docsearch-hit-color: var(--vp-c-text-2);--docsearch-hit-shadow: none}.DocSearch-Button{display:flex;justify-content:center;align-items:center;margin:0;padding:0;width:32px;height:55px;background:transparent;transition:border-color .25s}.DocSearch-Button:hover{background:transparent}.DocSearch-Button:focus{outline:1px dotted;outline:5px auto -webkit-focus-ring-color}.DocSearch-Button:focus:not(:focus-visible){outline:none!important}@media (min-width: 768px){.DocSearch-Button{justify-content:flex-start;border:1px solid transparent;border-radius:8px;padding:0 10px 0 12px;width:100%;height:40px;background-color:var(--vp-c-bg-alt)}.DocSearch-Button:hover{border-color:var(--vp-c-brand);background:var(--vp-c-bg-alt)}}.DocSearch-Button .DocSearch-Button-Container{display:flex;align-items:center}.DocSearch-Button .DocSearch-Search-Icon{position:relative;width:16px;height:16px;color:var(--vp-c-text-1);fill:currentColor;transition:color .5s}.DocSearch-Button:hover .DocSearch-Search-Icon{color:var(--vp-c-text-1)}@media (min-width: 768px){.DocSearch-Button .DocSearch-Search-Icon{top:1px;margin-right:8px;width:14px;height:14px;color:var(--vp-c-text-2)}}.DocSearch-Button .DocSearch-Button-Placeholder{display:none;margin-top:2px;padding:0 16px 0 0;font-size:13px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.DocSearch-Button:hover .DocSearch-Button-Placeholder{color:var(--vp-c-text-1)}@media (min-width: 768px){.DocSearch-Button .DocSearch-Button-Placeholder{display:inline-block}}.DocSearch-Button .DocSearch-Button-Keys{direction:ltr;display:none;min-width:auto}@media (min-width: 768px){.DocSearch-Button .DocSearch-Button-Keys{display:flex;align-items:center}}.DocSearch-Button .DocSearch-Button-Key{display:block;margin:2px 0 0;border:1px solid var(--vp-c-divider);border-right:none;border-radius:4px 0 0 4px;padding-left:6px;min-width:0;width:auto;height:22px;line-height:22px;font-family:var(--vp-font-family-base);font-size:12px;font-weight:500;transition:color .5s,border-color .5s}.DocSearch-Button .DocSearch-Button-Key+.DocSearch-Button-Key{border-right:1px solid var(--vp-c-divider);border-left:none;border-radius:0 4px 4px 0;padding-left:2px;padding-right:6px}.DocSearch-Button .DocSearch-Button-Key:first-child{font-size:1px;letter-spacing:-12px;color:transparent}.DocSearch-Button .DocSearch-Button-Key:first-child:after{content:var(--vp-meta-key);font-size:12px;letter-spacing:normal;color:var(--docsearch-muted-color)}.DocSearch-Button .DocSearch-Button-Key:first-child>*{display:none}.VPNavBarSearch{display:flex;align-items:center}@media (min-width: 768px){.VPNavBarSearch{flex-grow:1;padding-left:24px}}@media (min-width: 960px){.VPNavBarSearch{padding-left:32px}}.dark .DocSearch-Footer{border-top:1px solid var(--vp-c-divider)}.DocSearch-Form{border:1px solid var(--vp-c-brand);background-color:var(--vp-c-white)}.dark .DocSearch-Form{background-color:var(--vp-c-bg-soft-mute)}.DocSearch-Screen-Icon>svg{margin:auto}.icon[data-v-a72b6228]{display:inline-block;margin-top:-1px;margin-left:4px;width:11px;height:11px;fill:var(--vp-c-text-3);transition:fill .25s;flex-shrink:0}.VPNavBarMenuLink[data-v-fa5c4352]{display:flex;align-items:center;padding:0 12px;line-height:var(--vp-nav-height);font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.VPNavBarMenuLink.active[data-v-fa5c4352],.VPNavBarMenuLink[data-v-fa5c4352]:hover{color:var(--vp-c-brand)}.VPMenuGroup+.VPMenuLink[data-v-dcf4ab55]{margin:12px -12px 0;border-top:1px solid var(--vp-c-divider);padding:12px 12px 0}.link[data-v-dcf4ab55]{display:block;border-radius:6px;padding:0 12px;line-height:32px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);white-space:nowrap;transition:background-color .25s,color .25s}.link[data-v-dcf4ab55]:hover{color:var(--vp-c-brand);background-color:var(--vp-c-bg-elv-mute)}.link.active[data-v-dcf4ab55]{color:var(--vp-c-brand)}.VPMenuGroup[data-v-b48b6f40]{margin:12px -12px 0;border-top:1px solid var(--vp-c-divider);padding:12px 12px 0}.VPMenuGroup[data-v-b48b6f40]:first-child{margin-top:0;border-top:0;padding-top:0}.VPMenuGroup+.VPMenuGroup[data-v-b48b6f40]{margin-top:12px;border-top:1px solid var(--vp-c-divider)}.title[data-v-b48b6f40]{padding:0 12px;line-height:32px;font-size:14px;font-weight:600;color:var(--vp-c-text-2);white-space:nowrap;transition:color .25s}.VPMenu[data-v-ed414532]{border-radius:12px;padding:12px;min-width:128px;border:1px solid var(--vp-c-divider);background-color:var(--vp-c-bg-elv);box-shadow:var(--vp-shadow-3);transition:background-color .5s;max-height:calc(100vh - var(--vp-nav-height));overflow-y:auto}.VPMenu[data-v-ed414532] .group{margin:0 -12px;padding:0 12px 12px}.VPMenu[data-v-ed414532] .group+.group{border-top:1px solid var(--vp-c-divider);padding:11px 12px 12px}.VPMenu[data-v-ed414532] .group:last-child{padding-bottom:0}.VPMenu[data-v-ed414532] .group+.item{border-top:1px solid var(--vp-c-divider);padding:11px 16px 0}.VPMenu[data-v-ed414532] .item{padding:0 16px;white-space:nowrap}.VPMenu[data-v-ed414532] .label{flex-grow:1;line-height:28px;font-size:12px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.VPMenu[data-v-ed414532] .action{padding-left:24px}.VPFlyout[data-v-d146844d]{position:relative}.VPFlyout[data-v-d146844d]:hover{color:var(--vp-c-brand);transition:color .25s}.VPFlyout:hover .text[data-v-d146844d]{color:var(--vp-c-text-2)}.VPFlyout:hover .icon[data-v-d146844d]{fill:var(--vp-c-text-2)}.VPFlyout.active .text[data-v-d146844d]{color:var(--vp-c-brand)}.VPFlyout.active:hover .text[data-v-d146844d]{color:var(--vp-c-brand-dark)}.VPFlyout:hover .menu[data-v-d146844d],.button[aria-expanded=true]+.menu[data-v-d146844d]{opacity:1;visibility:visible;transform:translateY(0)}.button[data-v-d146844d]{display:flex;align-items:center;padding:0 12px;height:var(--vp-nav-height);color:var(--vp-c-text-1);transition:color .5s}.text[data-v-d146844d]{display:flex;align-items:center;line-height:var(--vp-nav-height);font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.option-icon[data-v-d146844d]{margin-right:0;width:16px;height:16px;fill:currentColor}.text-icon[data-v-d146844d]{margin-left:4px;width:14px;height:14px;fill:currentColor}.icon[data-v-d146844d]{width:20px;height:20px;fill:currentColor;transition:fill .25s}.menu[data-v-d146844d]{position:absolute;top:calc(var(--vp-nav-height) / 2 + 20px);right:0;opacity:0;visibility:hidden;transition:opacity .25s,visibility .25s,transform .25s}.VPNavBarMenu[data-v-a0e7a303]{display:none}@media (min-width: 768px){.VPNavBarMenu[data-v-a0e7a303]{display:flex}}.VPNavBarTranslations[data-v-9047dbea]{display:none}@media (min-width: 1280px){.VPNavBarTranslations[data-v-9047dbea]{display:flex;align-items:center}}.title[data-v-9047dbea]{padding:0 24px 0 12px;line-height:32px;font-size:14px;font-weight:700;color:var(--vp-c-text-1)}.VPSwitch[data-v-fb9fb2bc]{position:relative;border-radius:11px;display:block;width:40px;height:22px;flex-shrink:0;border:1px solid var(--vp-input-border-color);background-color:var(--vp-input-switch-bg-color);transition:border-color .25s}.VPSwitch[data-v-fb9fb2bc]:hover{border-color:var(--vp-input-hover-border-color)}.check[data-v-fb9fb2bc]{position:absolute;top:1px;left:1px;width:18px;height:18px;border-radius:50%;background-color:var(--vp-c-neutral-inverse);box-shadow:var(--vp-shadow-1);transition:transform .25s}.icon[data-v-fb9fb2bc]{position:relative;display:block;width:18px;height:18px;border-radius:50%;overflow:hidden}.icon[data-v-fb9fb2bc] svg{position:absolute;top:3px;left:3px;width:12px;height:12px;fill:var(--vp-c-text-2)}.dark .icon[data-v-fb9fb2bc] svg{fill:var(--vp-c-text-1);transition:opacity .25s}.sun[data-v-5314f25c]{opacity:1}.moon[data-v-5314f25c],.dark .sun[data-v-5314f25c]{opacity:0}.dark .moon[data-v-5314f25c]{opacity:1}.dark .VPSwitchAppearance[data-v-5314f25c] .check{transform:translate(18px)}.VPNavBarAppearance[data-v-ba9af2dd]{display:none}@media (min-width: 1280px){.VPNavBarAppearance[data-v-ba9af2dd]{display:flex;align-items:center}}.VPSocialLink[data-v-f4bd6624]{display:flex;justify-content:center;align-items:center;width:36px;height:36px;color:var(--vp-c-text-2);transition:color .5s}.VPSocialLink[data-v-f4bd6624]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPSocialLink[data-v-f4bd6624]>svg{width:20px;height:20px;fill:currentColor}.VPSocialLinks[data-v-fea619a9]{display:flex;flex-wrap:wrap;justify-content:center}.VPNavBarSocialLinks[data-v-49754fc9]{display:none}@media (min-width: 1280px){.VPNavBarSocialLinks[data-v-49754fc9]{display:flex;align-items:center}}.VPNavBarExtra[data-v-5ac0399f]{display:none;margin-right:-12px}@media (min-width: 768px){.VPNavBarExtra[data-v-5ac0399f]{display:block}}@media (min-width: 1280px){.VPNavBarExtra[data-v-5ac0399f]{display:none}}.trans-title[data-v-5ac0399f]{padding:0 24px 0 12px;line-height:32px;font-size:14px;font-weight:700;color:var(--vp-c-text-1)}.item.appearance[data-v-5ac0399f],.item.social-links[data-v-5ac0399f]{display:flex;align-items:center;padding:0 12px}.item.appearance[data-v-5ac0399f]{min-width:176px}.appearance-action[data-v-5ac0399f]{margin-right:-2px}.social-links-list[data-v-5ac0399f]{margin:-4px -8px}.VPNavBarHamburger[data-v-07c69bee]{display:flex;justify-content:center;align-items:center;width:48px;height:var(--vp-nav-height)}@media (min-width: 768px){.VPNavBarHamburger[data-v-07c69bee]{display:none}}.container[data-v-07c69bee]{position:relative;width:16px;height:14px;overflow:hidden}.VPNavBarHamburger:hover .top[data-v-07c69bee]{top:0;left:0;transform:translate(4px)}.VPNavBarHamburger:hover .middle[data-v-07c69bee]{top:6px;left:0;transform:translate(0)}.VPNavBarHamburger:hover .bottom[data-v-07c69bee]{top:12px;left:0;transform:translate(8px)}.VPNavBarHamburger.active .top[data-v-07c69bee]{top:6px;transform:translate(0) rotate(225deg)}.VPNavBarHamburger.active .middle[data-v-07c69bee]{top:6px;transform:translate(16px)}.VPNavBarHamburger.active .bottom[data-v-07c69bee]{top:6px;transform:translate(0) rotate(135deg)}.VPNavBarHamburger.active:hover .top[data-v-07c69bee],.VPNavBarHamburger.active:hover .middle[data-v-07c69bee],.VPNavBarHamburger.active:hover .bottom[data-v-07c69bee]{background-color:var(--vp-c-text-2);transition:top .25s,background-color .25s,transform .25s}.top[data-v-07c69bee],.middle[data-v-07c69bee],.bottom[data-v-07c69bee]{position:absolute;width:16px;height:2px;background-color:var(--vp-c-text-1);transition:top .25s,background-color .5s,transform .25s}.top[data-v-07c69bee]{top:0;left:0;transform:translate(0)}.middle[data-v-07c69bee]{top:6px;left:0;transform:translate(8px)}.bottom[data-v-07c69bee]{top:12px;left:0;transform:translate(4px)}.VPNavBar[data-v-528d5544]{position:relative;border-bottom:1px solid transparent;padding:0 8px 0 24px;height:var(--vp-nav-height);transition:border-color .5s,background-color .5s;pointer-events:none;white-space:nowrap}.VPNavBar.has-sidebar[data-v-528d5544]{border-bottom-color:var(--vp-c-gutter)}@media (min-width: 768px){.VPNavBar[data-v-528d5544]{padding:0 32px}}@media (min-width: 960px){.VPNavBar.has-sidebar[data-v-528d5544]{border-bottom-color:transparent;padding:0}.VPNavBar.fill[data-v-528d5544]:not(.has-sidebar){border-bottom-color:var(--vp-c-gutter);background-color:var(--vp-nav-bg-color)}}.container[data-v-528d5544]{display:flex;justify-content:space-between;margin:0 auto;max-width:calc(var(--vp-layout-max-width) - 64px);height:var(--vp-nav-height);pointer-events:none}.container>.title[data-v-528d5544],.container>.content[data-v-528d5544]{pointer-events:none}.container[data-v-528d5544] *{pointer-events:auto}@media (min-width: 960px){.VPNavBar.has-sidebar .container[data-v-528d5544]{max-width:100%}}.title[data-v-528d5544]{flex-shrink:0;height:calc(var(--vp-nav-height) - 1px);transition:background-color .5s}@media (min-width: 960px){.VPNavBar.has-sidebar .title[data-v-528d5544]{position:absolute;top:0;left:0;z-index:2;padding:0 32px;width:var(--vp-sidebar-width);height:var(--vp-nav-height);background-color:transparent}}@media (min-width: 1440px){.VPNavBar.has-sidebar .title[data-v-528d5544]{padding-left:max(32px,calc((100% - (var(--vp-layout-max-width) - 64px)) / 2));width:calc((100% - (var(--vp-layout-max-width) - 64px)) / 2 + var(--vp-sidebar-width) - 32px)}}.content[data-v-528d5544]{flex-grow:1}@media (min-width: 960px){.VPNavBar.has-sidebar .content[data-v-528d5544]{position:relative;z-index:1;padding-right:32px;padding-left:var(--vp-sidebar-width)}}@media (min-width: 1440px){.VPNavBar.has-sidebar .content[data-v-528d5544]{padding-right:calc((100vw - var(--vp-layout-max-width)) / 2 + 32px);padding-left:calc((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width))}}.content-body[data-v-528d5544]{display:flex;justify-content:flex-end;align-items:center;height:calc(var(--vp-nav-height) - 1px);transition:background-color .5s}@media (min-width: 960px){.VPNavBar.has-sidebar .content-body[data-v-528d5544],.VPNavBar.fill .content-body[data-v-528d5544]{position:relative;background-color:var(--vp-nav-bg-color)}}.menu+.translations[data-v-528d5544]:before,.menu+.appearance[data-v-528d5544]:before,.menu+.social-links[data-v-528d5544]:before,.translations+.appearance[data-v-528d5544]:before,.appearance+.social-links[data-v-528d5544]:before{margin-right:8px;margin-left:8px;width:1px;height:24px;background-color:var(--vp-c-divider);content:""}.menu+.appearance[data-v-528d5544]:before,.translations+.appearance[data-v-528d5544]:before{margin-right:16px}.appearance+.social-links[data-v-528d5544]:before{margin-left:16px}.social-links[data-v-528d5544]{margin-right:-8px}@media (min-width: 960px){.VPNavBar.has-sidebar .curtain[data-v-528d5544]{position:absolute;right:0;bottom:-31px;width:calc(100% - var(--vp-sidebar-width));height:32px}.VPNavBar.has-sidebar .curtain[data-v-528d5544]:before{display:block;width:100%;height:32px;background:linear-gradient(var(--vp-c-bg),transparent 70%);content:""}}@media (min-width: 1440px){.VPNavBar.has-sidebar .curtain[data-v-528d5544]{width:calc(100% - ((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width)))}}.VPNavScreenMenuLink[data-v-4a38c9c0]{display:block;border-bottom:1px solid var(--vp-c-divider);padding:12px 0 11px;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:border-color .25s,color .25s}.VPNavScreenMenuLink[data-v-4a38c9c0]:hover{color:var(--vp-c-brand)}.VPNavScreenMenuGroupLink[data-v-5b16e6cf]{display:block;margin-left:12px;line-height:32px;font-size:14px;font-weight:400;color:var(--vp-c-text-1);transition:color .25s}.VPNavScreenMenuGroupLink[data-v-5b16e6cf]:hover{color:var(--vp-c-brand)}.VPNavScreenMenuGroupSection[data-v-34625a7e]{display:block}.title[data-v-34625a7e]{line-height:32px;font-size:13px;font-weight:700;color:var(--vp-c-text-2);transition:color .25s}.VPNavScreenMenuGroup[data-v-a3dd94ad]{border-bottom:1px solid var(--vp-c-divider);height:48px;overflow:hidden;transition:border-color .5s}.VPNavScreenMenuGroup .items[data-v-a3dd94ad]{visibility:hidden}.VPNavScreenMenuGroup.open .items[data-v-a3dd94ad]{visibility:visible}.VPNavScreenMenuGroup.open[data-v-a3dd94ad]{padding-bottom:10px;height:auto}.VPNavScreenMenuGroup.open .button[data-v-a3dd94ad]{padding-bottom:6px;color:var(--vp-c-brand)}.VPNavScreenMenuGroup.open .button-icon[data-v-a3dd94ad]{transform:rotate(45deg)}.button[data-v-a3dd94ad]{display:flex;justify-content:space-between;align-items:center;padding:12px 4px 11px 0;width:100%;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-1);transition:color .25s}.button[data-v-a3dd94ad]:hover{color:var(--vp-c-brand)}.button-icon[data-v-a3dd94ad]{width:14px;height:14px;fill:var(--vp-c-text-2);transition:fill .5s,transform .25s}.group[data-v-a3dd94ad]:first-child{padding-top:0}.group+.group[data-v-a3dd94ad],.group+.item[data-v-a3dd94ad]{padding-top:4px}.VPNavScreenAppearance[data-v-cefb5854]{display:flex;justify-content:space-between;align-items:center;border-radius:8px;padding:12px 14px 12px 16px;background-color:var(--vp-c-bg-soft)}.text[data-v-cefb5854]{line-height:24px;font-size:12px;font-weight:500;color:var(--vp-c-text-2)}.VPNavScreenTranslations[data-v-4d5840eb]{height:24px;overflow:hidden}.VPNavScreenTranslations.open[data-v-4d5840eb]{height:auto}.title[data-v-4d5840eb]{display:flex;align-items:center;font-size:14px;font-weight:500;color:var(--vp-c-text-1)}.icon[data-v-4d5840eb]{width:16px;height:16px;fill:currentColor}.icon.lang[data-v-4d5840eb]{margin-right:8px}.icon.chevron[data-v-4d5840eb]{margin-left:4px}.list[data-v-4d5840eb]{padding:4px 0 0 24px}.link[data-v-4d5840eb]{line-height:32px;font-size:13px;color:var(--vp-c-text-1)}.VPNavScreen[data-v-8337e0f7]{position:fixed;top:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 1px);right:0;bottom:0;left:0;padding:0 32px;width:100%;background-color:var(--vp-nav-screen-bg-color);overflow-y:auto;transition:background-color .5s;pointer-events:auto}.VPNavScreen.fade-enter-active[data-v-8337e0f7],.VPNavScreen.fade-leave-active[data-v-8337e0f7]{transition:opacity .25s}.VPNavScreen.fade-enter-active .container[data-v-8337e0f7],.VPNavScreen.fade-leave-active .container[data-v-8337e0f7]{transition:transform .25s ease}.VPNavScreen.fade-enter-from[data-v-8337e0f7],.VPNavScreen.fade-leave-to[data-v-8337e0f7]{opacity:0}.VPNavScreen.fade-enter-from .container[data-v-8337e0f7],.VPNavScreen.fade-leave-to .container[data-v-8337e0f7]{transform:translateY(-8px)}@media (min-width: 768px){.VPNavScreen[data-v-8337e0f7]{display:none}}.container[data-v-8337e0f7]{margin:0 auto;padding:24px 0 96px;max-width:288px}.menu+.translations[data-v-8337e0f7],.menu+.appearance[data-v-8337e0f7],.translations+.appearance[data-v-8337e0f7]{margin-top:24px}.menu+.social-links[data-v-8337e0f7]{margin-top:16px}.appearance+.social-links[data-v-8337e0f7]{margin-top:16px}.VPNav[data-v-d58b459a]{position:relative;top:var(--vp-layout-top-height, 0px);left:0;z-index:var(--vp-z-index-nav);width:100%;pointer-events:none;transition:background-color .5s}@media (min-width: 960px){.VPNav[data-v-d58b459a]{position:fixed}}.root[data-v-65e3602e]{position:relative;z-index:1}.nested[data-v-65e3602e]{padding-left:13px}.outline-link[data-v-65e3602e]{display:block;line-height:28px;color:var(--vp-c-text-2);white-space:nowrap;overflow:hidden;text-overflow:ellipsis;transition:color .5s;font-weight:500}.outline-link[data-v-65e3602e]:hover,.outline-link.active[data-v-65e3602e]{color:var(--vp-c-text-1);transition:color .25s}.outline-link.nested[data-v-65e3602e]{padding-left:13px}.VPLocalNavOutlineDropdown[data-v-ba7b08b9]{padding:12px 20px 11px}.VPLocalNavOutlineDropdown button[data-v-ba7b08b9]{display:block;font-size:12px;font-weight:500;line-height:24px;color:var(--vp-c-text-2);transition:color .5s;position:relative}.VPLocalNavOutlineDropdown button[data-v-ba7b08b9]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPLocalNavOutlineDropdown button.open[data-v-ba7b08b9]{color:var(--vp-c-text-1)}.icon[data-v-ba7b08b9]{display:inline-block;vertical-align:middle;margin-left:2px;width:14px;height:14px;fill:currentColor}[data-v-ba7b08b9] .outline-link{font-size:14px;padding:2px 0}.open>.icon[data-v-ba7b08b9]{transform:rotate(90deg)}.items[data-v-ba7b08b9]{position:absolute;left:20px;right:20px;top:64px;background-color:var(--vp-local-nav-bg-color);padding:4px 10px 16px;border:1px solid var(--vp-c-divider);border-radius:8px;max-height:calc(var(--vp-vh, 100vh) - 86px);overflow:hidden auto;box-shadow:var(--vp-shadow-3)}.top-link[data-v-ba7b08b9]{display:block;color:var(--vp-c-brand);font-size:13px;font-weight:500;padding:6px 0;margin:0 13px 10px;border-bottom:1px solid var(--vp-c-divider)}.flyout-enter-active[data-v-ba7b08b9]{transition:all .2s ease-out}.flyout-leave-active[data-v-ba7b08b9]{transition:all .15s ease-in}.flyout-enter-from[data-v-ba7b08b9],.flyout-leave-to[data-v-ba7b08b9]{opacity:0;transform:translateY(-16px)}.VPLocalNav[data-v-8c9577da]{position:sticky;top:0;left:0;z-index:var(--vp-z-index-local-nav);display:flex;justify-content:space-between;align-items:center;border-bottom:1px solid var(--vp-c-gutter);padding-top:var(--vp-layout-top-height, 0px);width:100%;background-color:var(--vp-local-nav-bg-color);transition:border-color .5s,background-color .5s}@media (min-width: 960px){.VPLocalNav[data-v-8c9577da]{display:none}}.menu[data-v-8c9577da]{display:flex;align-items:center;padding:12px 24px 11px;line-height:24px;font-size:12px;font-weight:500;color:var(--vp-c-text-2);transition:color .5s}.menu[data-v-8c9577da]:hover{color:var(--vp-c-text-1);transition:color .25s}@media (min-width: 768px){.menu[data-v-8c9577da]{padding:0 32px}}.menu-icon[data-v-8c9577da]{margin-right:8px;width:16px;height:16px;fill:currentColor}.VPOutlineDropdown[data-v-8c9577da]{padding:12px 24px 11px}@media (min-width: 768px){.VPOutlineDropdown[data-v-8c9577da]{padding:12px 32px 11px}}.VPSidebarItem.level-0[data-v-a2d5c4ca]{padding-bottom:24px}.VPSidebarItem.collapsed.level-0[data-v-a2d5c4ca]{padding-bottom:10px}.item[data-v-a2d5c4ca]{position:relative;display:flex;width:100%}.VPSidebarItem.collapsible>.item[data-v-a2d5c4ca]{cursor:pointer}.indicator[data-v-a2d5c4ca]{position:absolute;top:6px;bottom:6px;left:-17px;width:1px;transition:background-color .25s}.VPSidebarItem.level-2.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-active>.item>.indicator[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-active>.item>.indicator[data-v-a2d5c4ca]{background-color:var(--vp-c-brand)}.link[data-v-a2d5c4ca]{display:flex;align-items:center;flex-grow:1}.text[data-v-a2d5c4ca]{flex-grow:1;padding:4px 0;line-height:24px;font-size:14px;transition:color .25s}.VPSidebarItem.level-0 .text[data-v-a2d5c4ca]{font-weight:700;color:var(--vp-c-text-1)}.VPSidebarItem.level-1 .text[data-v-a2d5c4ca],.VPSidebarItem.level-2 .text[data-v-a2d5c4ca],.VPSidebarItem.level-3 .text[data-v-a2d5c4ca],.VPSidebarItem.level-4 .text[data-v-a2d5c4ca],.VPSidebarItem.level-5 .text[data-v-a2d5c4ca]{font-weight:500;color:var(--vp-c-text-2)}.VPSidebarItem.level-0.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-1.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-2.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-link>.item>.link:hover .text[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-link>.item>.link:hover .text[data-v-a2d5c4ca]{color:var(--vp-c-brand)}.VPSidebarItem.level-0.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-1.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-2.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-3.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-4.has-active>.item>.link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-5.has-active>.item>.link>.text[data-v-a2d5c4ca]{color:var(--vp-c-text-1)}.VPSidebarItem.level-0.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-1.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-2.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-3.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-4.is-active>.item .link>.text[data-v-a2d5c4ca],.VPSidebarItem.level-5.is-active>.item .link>.text[data-v-a2d5c4ca]{color:var(--vp-c-brand)}.caret[data-v-a2d5c4ca]{display:flex;justify-content:center;align-items:center;margin-right:-7px;width:32px;height:32px;color:var(--vp-c-text-3);cursor:pointer;transition:color .25s}.item:hover .caret[data-v-a2d5c4ca]{color:var(--vp-c-text-2)}.item:hover .caret[data-v-a2d5c4ca]:hover{color:var(--vp-c-text-1)}.caret-icon[data-v-a2d5c4ca]{width:18px;height:18px;fill:currentColor;transform:rotate(90deg);transition:transform .25s}.VPSidebarItem.collapsed .caret-icon[data-v-a2d5c4ca]{transform:rotate(0)}.VPSidebarItem.level-1 .items[data-v-a2d5c4ca],.VPSidebarItem.level-2 .items[data-v-a2d5c4ca],.VPSidebarItem.level-3 .items[data-v-a2d5c4ca],.VPSidebarItem.level-4 .items[data-v-a2d5c4ca],.VPSidebarItem.level-5 .items[data-v-a2d5c4ca]{border-left:1px solid var(--vp-c-divider);padding-left:16px}.VPSidebarItem.collapsed .items[data-v-a2d5c4ca]{display:none}.VPSidebar[data-v-c7a4f3fe]{position:fixed;top:var(--vp-layout-top-height, 0px);bottom:0;left:0;z-index:var(--vp-z-index-sidebar);padding:32px 32px 96px;width:calc(100vw - 64px);max-width:320px;background-color:var(--vp-sidebar-bg-color);opacity:0;box-shadow:var(--vp-c-shadow-3);overflow-x:hidden;overflow-y:auto;transform:translate(-100%);transition:opacity .5s,transform .25s ease;overscroll-behavior:contain}.VPSidebar.open[data-v-c7a4f3fe]{opacity:1;visibility:visible;transform:translate(0);transition:opacity .25s,transform .5s cubic-bezier(.19,1,.22,1)}.dark .VPSidebar[data-v-c7a4f3fe]{box-shadow:var(--vp-shadow-1)}@media (min-width: 960px){.VPSidebar[data-v-c7a4f3fe]{z-index:1;padding-top:var(--vp-nav-height);padding-bottom:128px;width:var(--vp-sidebar-width);max-width:100%;background-color:var(--vp-sidebar-bg-color);opacity:1;visibility:visible;box-shadow:none;transform:translate(0)}}@media (min-width: 1440px){.VPSidebar[data-v-c7a4f3fe]{padding-left:max(32px,calc((100% - (var(--vp-layout-max-width) - 64px)) / 2));width:calc((100% - (var(--vp-layout-max-width) - 64px)) / 2 + var(--vp-sidebar-width) - 32px)}}@media (min-width: 960px){.curtain[data-v-c7a4f3fe]{position:sticky;top:-64px;left:0;z-index:1;margin-top:calc(var(--vp-nav-height) * -1);margin-right:-32px;margin-left:-32px;height:var(--vp-nav-height);background-color:var(--vp-sidebar-bg-color)}}.nav[data-v-c7a4f3fe]{outline:0}.group+.group[data-v-c7a4f3fe]{border-top:1px solid var(--vp-c-divider);padding-top:10px}@media (min-width: 960px){.group[data-v-c7a4f3fe]{padding-top:10px;width:calc(var(--vp-sidebar-width) - 64px)}}.VPButton[data-v-30800ca8]{display:inline-block;border:1px solid transparent;text-align:center;font-weight:600;white-space:nowrap;transition:color .25s,border-color .25s,background-color .25s}.VPButton[data-v-30800ca8]:active{transition:color .1s,border-color .1s,background-color .1s}.VPButton.medium[data-v-30800ca8]{border-radius:20px;padding:0 20px;line-height:38px;font-size:14px}.VPButton.big[data-v-30800ca8]{border-radius:24px;padding:0 24px;line-height:46px;font-size:16px}.VPButton.brand[data-v-30800ca8]{border-color:var(--vp-button-brand-border);color:var(--vp-button-brand-text);background-color:var(--vp-button-brand-bg)}.VPButton.brand[data-v-30800ca8]:hover{border-color:var(--vp-button-brand-hover-border);color:var(--vp-button-brand-hover-text);background-color:var(--vp-button-brand-hover-bg)}.VPButton.brand[data-v-30800ca8]:active{border-color:var(--vp-button-brand-active-border);color:var(--vp-button-brand-active-text);background-color:var(--vp-button-brand-active-bg)}.VPButton.alt[data-v-30800ca8]{border-color:var(--vp-button-alt-border);color:var(--vp-button-alt-text);background-color:var(--vp-button-alt-bg)}.VPButton.alt[data-v-30800ca8]:hover{border-color:var(--vp-button-alt-hover-border);color:var(--vp-button-alt-hover-text);background-color:var(--vp-button-alt-hover-bg)}.VPButton.alt[data-v-30800ca8]:active{border-color:var(--vp-button-alt-active-border);color:var(--vp-button-alt-active-text);background-color:var(--vp-button-alt-active-bg)}.VPButton.sponsor[data-v-30800ca8]{border-color:var(--vp-button-sponsor-border);color:var(--vp-button-sponsor-text);background-color:var(--vp-button-sponsor-bg)}.VPButton.sponsor[data-v-30800ca8]:hover{border-color:var(--vp-button-sponsor-hover-border);color:var(--vp-button-sponsor-hover-text);background-color:var(--vp-button-sponsor-hover-bg)}.VPButton.sponsor[data-v-30800ca8]:active{border-color:var(--vp-button-sponsor-active-border);color:var(--vp-button-sponsor-active-text);background-color:var(--vp-button-sponsor-active-bg)}.VPHero[data-v-9e7a9339]{margin-top:calc((var(--vp-nav-height) + var(--vp-layout-top-height, 0px)) * -1);padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 48px) 24px 48px}@media (min-width: 640px){.VPHero[data-v-9e7a9339]{padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 80px) 48px 64px}}@media (min-width: 960px){.VPHero[data-v-9e7a9339]{padding:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 80px) 64px 64px}}.container[data-v-9e7a9339]{display:flex;flex-direction:column;margin:0 auto;max-width:1152px}@media (min-width: 960px){.container[data-v-9e7a9339]{flex-direction:row}}.main[data-v-9e7a9339]{position:relative;z-index:10;order:2;flex-grow:1;flex-shrink:0}.VPHero.has-image .container[data-v-9e7a9339]{text-align:center}@media (min-width: 960px){.VPHero.has-image .container[data-v-9e7a9339]{text-align:left}}@media (min-width: 960px){.main[data-v-9e7a9339]{order:1;width:calc((100% / 3) * 2)}.VPHero.has-image .main[data-v-9e7a9339]{max-width:592px}}.name[data-v-9e7a9339],.text[data-v-9e7a9339]{max-width:392px;letter-spacing:-.4px;line-height:40px;font-size:32px;font-weight:700;white-space:pre-wrap}.VPHero.has-image .name[data-v-9e7a9339],.VPHero.has-image .text[data-v-9e7a9339]{margin:0 auto}.name[data-v-9e7a9339]{color:var(--vp-home-hero-name-color)}.clip[data-v-9e7a9339]{background:var(--vp-home-hero-name-background);-webkit-background-clip:text;background-clip:text;-webkit-text-fill-color:var(--vp-home-hero-name-color)}@media (min-width: 640px){.name[data-v-9e7a9339],.text[data-v-9e7a9339]{max-width:576px;line-height:56px;font-size:48px}}@media (min-width: 960px){.name[data-v-9e7a9339],.text[data-v-9e7a9339]{line-height:64px;font-size:56px}.VPHero.has-image .name[data-v-9e7a9339],.VPHero.has-image .text[data-v-9e7a9339]{margin:0}}.tagline[data-v-9e7a9339]{padding-top:8px;max-width:392px;line-height:28px;font-size:18px;font-weight:500;white-space:pre-wrap;color:var(--vp-c-text-2)}.VPHero.has-image .tagline[data-v-9e7a9339]{margin:0 auto}@media (min-width: 640px){.tagline[data-v-9e7a9339]{padding-top:12px;max-width:576px;line-height:32px;font-size:20px}}@media (min-width: 960px){.tagline[data-v-9e7a9339]{line-height:36px;font-size:24px}.VPHero.has-image .tagline[data-v-9e7a9339]{margin:0}}.actions[data-v-9e7a9339]{display:flex;flex-wrap:wrap;margin:-6px;padding-top:24px}.VPHero.has-image .actions[data-v-9e7a9339]{justify-content:center}@media (min-width: 640px){.actions[data-v-9e7a9339]{padding-top:32px}}@media (min-width: 960px){.VPHero.has-image .actions[data-v-9e7a9339]{justify-content:flex-start}}.action[data-v-9e7a9339]{flex-shrink:0;padding:6px}.image[data-v-9e7a9339]{order:1;margin:-76px -24px -48px}@media (min-width: 640px){.image[data-v-9e7a9339]{margin:-108px -24px -48px}}@media (min-width: 960px){.image[data-v-9e7a9339]{flex-grow:1;order:2;margin:0;min-height:100%}}.image-container[data-v-9e7a9339]{position:relative;margin:0 auto;width:320px;height:320px}@media (min-width: 640px){.image-container[data-v-9e7a9339]{width:392px;height:392px}}@media (min-width: 960px){.image-container[data-v-9e7a9339]{display:flex;justify-content:center;align-items:center;width:100%;height:100%;transform:translate(-32px,-32px)}}.image-bg[data-v-9e7a9339]{position:absolute;top:50%;left:50%;border-radius:50%;width:192px;height:192px;background-image:var(--vp-home-hero-image-background-image);filter:var(--vp-home-hero-image-filter);transform:translate(-50%,-50%)}@media (min-width: 640px){.image-bg[data-v-9e7a9339]{width:256px;height:256px}}@media (min-width: 960px){.image-bg[data-v-9e7a9339]{width:320px;height:320px}}[data-v-9e7a9339] .image-src{position:absolute;top:50%;left:50%;max-width:192px;max-height:192px;transform:translate(-50%,-50%)}@media (min-width: 640px){[data-v-9e7a9339] .image-src{max-width:256px;max-height:256px}}@media (min-width: 960px){[data-v-9e7a9339] .image-src{max-width:320px;max-height:320px}}.VPFeature[data-v-c4ab7714]{display:block;border:1px solid var(--vp-c-bg-soft);border-radius:12px;height:100%;background-color:var(--vp-c-bg-soft);transition:border-color .25s,background-color .25s}.VPFeature.link[data-v-c4ab7714]:hover{border-color:var(--vp-c-brand);background-color:var(--vp-c-bg-soft-up)}.box[data-v-c4ab7714]{display:flex;flex-direction:column;padding:24px;height:100%}.VPFeature[data-v-c4ab7714] .VPImage{width:48px;height:48px;margin-bottom:20px}.icon[data-v-c4ab7714]{display:flex;justify-content:center;align-items:center;margin-bottom:20px;border-radius:6px;background-color:var(--vp-c-bg-soft-down);width:48px;height:48px;font-size:24px;transition:background-color .25s}.title[data-v-c4ab7714]{line-height:24px;font-size:16px;font-weight:600}.details[data-v-c4ab7714]{flex-grow:1;padding-top:8px;line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.link-text[data-v-c4ab7714]{padding-top:8px}.link-text-value[data-v-c4ab7714]{display:flex;align-items:center;font-size:14px;font-weight:500;color:var(--vp-c-brand)}.link-text-icon[data-v-c4ab7714]{display:inline-block;margin-left:6px;width:14px;height:14px;fill:currentColor}.VPFeatures[data-v-1cc0eb36]{position:relative;padding:0 24px}@media (min-width: 640px){.VPFeatures[data-v-1cc0eb36]{padding:0 48px}}@media (min-width: 960px){.VPFeatures[data-v-1cc0eb36]{padding:0 64px}}.container[data-v-1cc0eb36]{margin:0 auto;max-width:1152px}.items[data-v-1cc0eb36]{display:flex;flex-wrap:wrap;margin:-8px}.item[data-v-1cc0eb36]{padding:8px;width:100%}@media (min-width: 640px){.item.grid-2[data-v-1cc0eb36],.item.grid-4[data-v-1cc0eb36],.item.grid-6[data-v-1cc0eb36]{width:50%}}@media (min-width: 768px){.item.grid-2[data-v-1cc0eb36],.item.grid-4[data-v-1cc0eb36]{width:50%}.item.grid-3[data-v-1cc0eb36],.item.grid-6[data-v-1cc0eb36]{width:calc(100% / 3)}}@media (min-width: 960px){.item.grid-4[data-v-1cc0eb36]{width:25%}}.VPHome[data-v-0cc58ac0]{padding-bottom:96px}.VPHome[data-v-0cc58ac0] .VPHomeSponsors{margin-top:112px;margin-bottom:-128px}@media (min-width: 768px){.VPHome[data-v-0cc58ac0]{padding-bottom:128px}}.VPDocAsideOutline[data-v-7c84bbba]{display:none}.VPDocAsideOutline.has-outline[data-v-7c84bbba]{display:block}.content[data-v-7c84bbba]{position:relative;border-left:1px solid var(--vp-c-divider);padding-left:16px;font-size:13px;font-weight:500}.outline-marker[data-v-7c84bbba]{position:absolute;top:32px;left:-1px;z-index:0;opacity:0;width:1px;height:18px;background-color:var(--vp-c-brand);transition:top .25s cubic-bezier(0,1,.5,1),background-color .5s,opacity .25s}.outline-title[data-v-7c84bbba]{letter-spacing:.4px;line-height:28px;font-size:13px;font-weight:600}.VPDocAside[data-v-ac259e18]{display:flex;flex-direction:column;flex-grow:1}.spacer[data-v-ac259e18]{flex-grow:1}.VPDocAside[data-v-ac259e18] .spacer+.VPDocAsideSponsors,.VPDocAside[data-v-ac259e18] .spacer+.VPDocAsideCarbonAds{margin-top:24px}.VPDocAside[data-v-ac259e18] .VPDocAsideSponsors+.VPDocAsideCarbonAds{margin-top:16px}.VPLastUpdated[data-v-d35b8cb4]{line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}@media (min-width: 640px){.VPLastUpdated[data-v-d35b8cb4]{line-height:32px;font-size:14px;font-weight:500}}.VPDocFooter[data-v-2b4f87e4]{margin-top:64px}.edit-info[data-v-2b4f87e4]{padding-bottom:18px}@media (min-width: 640px){.edit-info[data-v-2b4f87e4]{display:flex;justify-content:space-between;align-items:center;padding-bottom:14px}}.edit-link-button[data-v-2b4f87e4]{display:flex;align-items:center;border:0;line-height:32px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:color .25s}.edit-link-button[data-v-2b4f87e4]:hover{color:var(--vp-c-brand-dark)}.edit-link-icon[data-v-2b4f87e4]{margin-right:8px;width:14px;height:14px;fill:currentColor}.prev-next[data-v-2b4f87e4]{border-top:1px solid var(--vp-c-divider);padding-top:24px}@media (min-width: 640px){.prev-next[data-v-2b4f87e4]{display:flex}}.pager.has-prev[data-v-2b4f87e4]{padding-top:8px}@media (min-width: 640px){.pager[data-v-2b4f87e4]{display:flex;flex-direction:column;flex-shrink:0;width:50%}.pager.has-prev[data-v-2b4f87e4]{padding-top:0;padding-left:16px}}.pager-link[data-v-2b4f87e4]{display:block;border:1px solid var(--vp-c-divider);border-radius:8px;padding:11px 16px 13px;width:100%;height:100%;transition:border-color .25s}.pager-link[data-v-2b4f87e4]:hover{border-color:var(--vp-c-brand)}.pager-link.next[data-v-2b4f87e4]{margin-left:auto;text-align:right}.desc[data-v-2b4f87e4]{display:block;line-height:20px;font-size:12px;font-weight:500;color:var(--vp-c-text-2)}.title[data-v-2b4f87e4]{display:block;line-height:20px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:color .25s}.VPDocOutlineDropdown[data-v-7fff1eb3]{margin-bottom:42px}.VPDocOutlineDropdown button[data-v-7fff1eb3]{display:block;font-size:14px;font-weight:500;line-height:24px;color:var(--vp-c-text-2);transition:color .5s;border:1px solid var(--vp-c-border);padding:4px 12px;border-radius:8px}.VPDocOutlineDropdown button[data-v-7fff1eb3]:hover{color:var(--vp-c-text-1);transition:color .25s}.VPDocOutlineDropdown button.open[data-v-7fff1eb3]{color:var(--vp-c-text-1)}.icon[data-v-7fff1eb3]{display:inline-block;vertical-align:middle;margin-left:2px;width:14px;height:14px;fill:currentColor}[data-v-7fff1eb3] .outline-link{font-size:13px}.open>.icon[data-v-7fff1eb3]{transform:rotate(90deg)}.items[data-v-7fff1eb3]{margin-top:10px;border-left:1px solid var(--vp-c-divider)}.VPDoc[data-v-4c07d6a3]{padding:32px 24px 96px;width:100%}.VPDoc .VPDocOutlineDropdown[data-v-4c07d6a3]{display:none}@media (min-width: 960px) and (max-width: 1280px){.VPDoc .VPDocOutlineDropdown[data-v-4c07d6a3]{display:block}}@media (min-width: 768px){.VPDoc[data-v-4c07d6a3]{padding:48px 32px 128px}}@media (min-width: 960px){.VPDoc[data-v-4c07d6a3]{padding:32px 32px 0}.VPDoc:not(.has-sidebar) .container[data-v-4c07d6a3]{display:flex;justify-content:center;max-width:992px}.VPDoc:not(.has-sidebar) .content[data-v-4c07d6a3]{max-width:752px}}@media (min-width: 1280px){.VPDoc .container[data-v-4c07d6a3]{display:flex;justify-content:center}.VPDoc .aside[data-v-4c07d6a3]{display:block}}@media (min-width: 1440px){.VPDoc:not(.has-sidebar) .content[data-v-4c07d6a3]{max-width:784px}.VPDoc:not(.has-sidebar) .container[data-v-4c07d6a3]{max-width:1104px}}.container[data-v-4c07d6a3]{margin:0 auto;width:100%}.aside[data-v-4c07d6a3]{position:relative;display:none;order:2;flex-grow:1;padding-left:32px;width:100%;max-width:256px}.left-aside[data-v-4c07d6a3]{order:1;padding-left:unset;padding-right:32px}.aside-container[data-v-4c07d6a3]{position:fixed;top:0;padding-top:calc(var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + var(--vp-doc-top-height, 0px) + 32px);width:224px;height:100vh;overflow-x:hidden;overflow-y:auto;scrollbar-width:none}.aside-container[data-v-4c07d6a3]::-webkit-scrollbar{display:none}.aside-curtain[data-v-4c07d6a3]{position:fixed;bottom:0;z-index:10;width:224px;height:32px;background:linear-gradient(transparent,var(--vp-c-bg) 70%)}.aside-content[data-v-4c07d6a3]{display:flex;flex-direction:column;min-height:calc(100vh - (var(--vp-nav-height) + var(--vp-layout-top-height, 0px) + 32px));padding-bottom:32px}.content[data-v-4c07d6a3]{position:relative;margin:0 auto;width:100%}@media (min-width: 960px){.content[data-v-4c07d6a3]{padding:0 32px 128px}}@media (min-width: 1280px){.content[data-v-4c07d6a3]{order:1;margin:0;min-width:640px}}.content-container[data-v-4c07d6a3]{margin:0 auto}.VPDoc.has-aside .content-container[data-v-4c07d6a3]{max-width:688px}.NotFound[data-v-135d8cce]{padding:64px 24px 96px;text-align:center}@media (min-width: 768px){.NotFound[data-v-135d8cce]{padding:96px 32px 168px}}.code[data-v-135d8cce]{line-height:64px;font-size:64px;font-weight:600}.title[data-v-135d8cce]{padding-top:12px;letter-spacing:2px;line-height:20px;font-size:20px;font-weight:700}.divider[data-v-135d8cce]{margin:24px auto 18px;width:64px;height:1px;background-color:var(--vp-c-divider)}.quote[data-v-135d8cce]{margin:0 auto;max-width:256px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.action[data-v-135d8cce]{padding-top:20px}.link[data-v-135d8cce]{display:inline-block;border:1px solid var(--vp-c-brand);border-radius:16px;padding:3px 16px;font-size:14px;font-weight:500;color:var(--vp-c-brand);transition:border-color .25s,color .25s}.link[data-v-135d8cce]:hover{border-color:var(--vp-c-brand-dark);color:var(--vp-c-brand-dark)}.VPContent[data-v-185f5406]{flex-grow:1;flex-shrink:0;margin:var(--vp-layout-top-height, 0px) auto 0;width:100%}.VPContent.is-home[data-v-185f5406]{width:100%;max-width:100%}.VPContent.has-sidebar[data-v-185f5406]{margin:0}@media (min-width: 960px){.VPContent[data-v-185f5406]{padding-top:var(--vp-nav-height)}.VPContent.has-sidebar[data-v-185f5406]{margin:var(--vp-layout-top-height, 0px) 0 0;padding-left:var(--vp-sidebar-width)}}@media (min-width: 1440px){.VPContent.has-sidebar[data-v-185f5406]{padding-right:calc((100vw - var(--vp-layout-max-width)) / 2);padding-left:calc((100vw - var(--vp-layout-max-width)) / 2 + var(--vp-sidebar-width))}}.VPFooter[data-v-e35069ff]{position:relative;z-index:var(--vp-z-index-footer);border-top:1px solid var(--vp-c-gutter);padding:32px 24px;background-color:var(--vp-c-bg)}.VPFooter.has-sidebar[data-v-e35069ff]{display:none}@media (min-width: 768px){.VPFooter[data-v-e35069ff]{padding:32px}}.container[data-v-e35069ff]{margin:0 auto;max-width:var(--vp-layout-max-width);text-align:center}.message[data-v-e35069ff],.copyright[data-v-e35069ff]{line-height:24px;font-size:14px;font-weight:500;color:var(--vp-c-text-2)}.message[data-v-e35069ff]{order:2}.copyright[data-v-e35069ff]{order:1}.Layout[data-v-b4389d08]{display:flex;flex-direction:column;min-height:100vh}.VPHomeSponsors[data-v-c6a38a06]{border-top:1px solid var(--vp-c-gutter);padding:88px 24px 96px;background-color:var(--vp-c-bg)}.container[data-v-c6a38a06]{margin:0 auto;max-width:1152px}.love[data-v-c6a38a06]{margin:0 auto;width:28px;height:28px;color:var(--vp-c-text-3)}.icon[data-v-c6a38a06]{width:28px;height:28px;fill:currentColor}.message[data-v-c6a38a06]{margin:0 auto;padding-top:10px;max-width:320px;text-align:center;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}.sponsors[data-v-c6a38a06]{padding-top:32px}.action[data-v-c6a38a06]{padding-top:40px;text-align:center}.VPTeamPage[data-v-77c11d23]{padding-bottom:96px}@media (min-width: 768px){.VPTeamPage[data-v-77c11d23]{padding-bottom:128px}}.VPTeamPageSection+.VPTeamPageSection[data-v-77c11d23-s],.VPTeamMembers+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:64px}.VPTeamMembers+.VPTeamMembers[data-v-77c11d23-s]{margin-top:24px}@media (min-width: 768px){.VPTeamPageTitle+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:16px}.VPTeamPageSection+.VPTeamPageSection[data-v-77c11d23-s],.VPTeamMembers+.VPTeamPageSection[data-v-77c11d23-s]{margin-top:96px}}.VPTeamMembers[data-v-77c11d23-s]{padding:0 24px}@media (min-width: 768px){.VPTeamMembers[data-v-77c11d23-s]{padding:0 48px}}@media (min-width: 960px){.VPTeamMembers[data-v-77c11d23-s]{padding:0 64px}}.VPTeamPageTitle[data-v-9bd857d2]{padding:48px 32px;text-align:center}@media (min-width: 768px){.VPTeamPageTitle[data-v-9bd857d2]{padding:64px 48px 48px}}@media (min-width: 960px){.VPTeamPageTitle[data-v-9bd857d2]{padding:80px 64px 48px}}.title[data-v-9bd857d2]{letter-spacing:0;line-height:44px;font-size:36px;font-weight:500}@media (min-width: 768px){.title[data-v-9bd857d2]{letter-spacing:-.5px;line-height:56px;font-size:48px}}.lead[data-v-9bd857d2]{margin:0 auto;max-width:512px;padding-top:12px;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}@media (min-width: 768px){.lead[data-v-9bd857d2]{max-width:592px;letter-spacing:.15px;line-height:28px;font-size:20px}}.VPTeamPageSection[data-v-ed31233b]{padding:0 32px}@media (min-width: 768px){.VPTeamPageSection[data-v-ed31233b]{padding:0 48px}}@media (min-width: 960px){.VPTeamPageSection[data-v-ed31233b]{padding:0 64px}}.title[data-v-ed31233b]{position:relative;margin:0 auto;max-width:1152px;text-align:center;color:var(--vp-c-text-2)}.title-line[data-v-ed31233b]{position:absolute;top:16px;left:0;width:100%;height:1px;background-color:var(--vp-c-divider)}.title-text[data-v-ed31233b]{position:relative;display:inline-block;padding:0 24px;letter-spacing:0;line-height:32px;font-size:20px;font-weight:500;background-color:var(--vp-c-bg)}.lead[data-v-ed31233b]{margin:0 auto;max-width:480px;padding-top:12px;text-align:center;line-height:24px;font-size:16px;font-weight:500;color:var(--vp-c-text-2)}.members[data-v-ed31233b]{padding-top:40px}.VPTeamMembersItem[data-v-2c315693]{display:flex;flex-direction:column;gap:2px;border-radius:12px;width:100%;height:100%;overflow:hidden}.VPTeamMembersItem.small .profile[data-v-2c315693]{padding:32px}.VPTeamMembersItem.small .data[data-v-2c315693]{padding-top:20px}.VPTeamMembersItem.small .avatar[data-v-2c315693]{width:64px;height:64px}.VPTeamMembersItem.small .name[data-v-2c315693]{line-height:24px;font-size:16px}.VPTeamMembersItem.small .affiliation[data-v-2c315693]{padding-top:4px;line-height:20px;font-size:14px}.VPTeamMembersItem.small .desc[data-v-2c315693]{padding-top:12px;line-height:20px;font-size:14px}.VPTeamMembersItem.small .links[data-v-2c315693]{margin:0 -16px -20px;padding:10px 0 0}.VPTeamMembersItem.medium .profile[data-v-2c315693]{padding:48px 32px}.VPTeamMembersItem.medium .data[data-v-2c315693]{padding-top:24px;text-align:center}.VPTeamMembersItem.medium .avatar[data-v-2c315693]{width:96px;height:96px}.VPTeamMembersItem.medium .name[data-v-2c315693]{letter-spacing:.15px;line-height:28px;font-size:20px}.VPTeamMembersItem.medium .affiliation[data-v-2c315693]{padding-top:4px;font-size:16px}.VPTeamMembersItem.medium .desc[data-v-2c315693]{padding-top:16px;max-width:288px;font-size:16px}.VPTeamMembersItem.medium .links[data-v-2c315693]{margin:0 -16px -12px;padding:16px 12px 0}.profile[data-v-2c315693]{flex-grow:1;background-color:var(--vp-c-bg-soft)}.data[data-v-2c315693]{text-align:center}.avatar[data-v-2c315693]{position:relative;flex-shrink:0;margin:0 auto;border-radius:50%;box-shadow:var(--vp-shadow-3)}.avatar-img[data-v-2c315693]{position:absolute;top:0;right:0;bottom:0;left:0;border-radius:50%;object-fit:cover}.name[data-v-2c315693]{margin:0;font-weight:600}.affiliation[data-v-2c315693]{margin:0;font-weight:500;color:var(--vp-c-text-2)}.org.link[data-v-2c315693]{color:var(--vp-c-text-2);transition:color .25s}.org.link[data-v-2c315693]:hover{color:var(--vp-c-brand)}.desc[data-v-2c315693]{margin:0 auto}.desc[data-v-2c315693] a{font-weight:500;color:var(--vp-c-brand);text-decoration-style:dotted;transition:color .25s}.links[data-v-2c315693]{display:flex;justify-content:center;height:56px}.sp-link[data-v-2c315693]{display:flex;justify-content:center;align-items:center;text-align:center;padding:16px;font-size:14px;font-weight:500;color:var(--vp-c-sponsor);background-color:var(--vp-c-bg-soft);transition:color .25s,background-color .25s}.sp .sp-link.link[data-v-2c315693]:hover,.sp .sp-link.link[data-v-2c315693]:focus{outline:none;color:var(--vp-c-white);background-color:var(--vp-c-sponsor)}.sp-icon[data-v-2c315693]{margin-right:8px;width:16px;height:16px;fill:currentColor}.VPTeamMembers.small .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(224px,1fr))}.VPTeamMembers.small.count-1 .container[data-v-9fb4c5f4]{max-width:276px}.VPTeamMembers.small.count-2 .container[data-v-9fb4c5f4]{max-width:576px}.VPTeamMembers.small.count-3 .container[data-v-9fb4c5f4]{max-width:876px}.VPTeamMembers.medium .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(256px,1fr))}@media (min-width: 375px){.VPTeamMembers.medium .container[data-v-9fb4c5f4]{grid-template-columns:repeat(auto-fit,minmax(288px,1fr))}}.VPTeamMembers.medium.count-1 .container[data-v-9fb4c5f4]{max-width:368px}.VPTeamMembers.medium.count-2 .container[data-v-9fb4c5f4]{max-width:760px}.container[data-v-9fb4c5f4]{display:grid;gap:24px;margin:0 auto;max-width:1152px}.m-home-layout .image-src:hover{transform:translate(-50%,-50%) rotate(666turn);transition:transform 59s 1s cubic-bezier(.3,0,.8,1)}.m-home-layout .details small{opacity:.8}.m-home-layout .item:last-child .details{display:flex;justify-content:flex-end;align-items:end}#nprogress{pointer-events:none}#nprogress .bar{background:var(--vp-c-brand);position:fixed;z-index:1031;top:0;left:0;width:100%;height:2px}#nprogress .peg{display:block;position:absolute;right:0px;width:100px;height:100%;box-shadow:0 0 10px var(--vp-c-brand),0 0 5px var(--vp-c-brand);opacity:1;-webkit-transform:rotate(3deg) translate(0px,-4px);-ms-transform:rotate(3deg) translate(0px,-4px);transform:rotate(3deg) translateY(-4px)}#nprogress .spinner{display:block;position:fixed;z-index:1031;top:15px;right:15px}#nprogress .spinner-icon{width:18px;height:18px;box-sizing:border-box;border:solid 2px transparent;border-top-color:var(--vp-c-brand);border-left-color:var(--vp-c-brand);border-radius:50%;-webkit-animation:nprogress-spinner .4s linear infinite;animation:nprogress-spinner .4s linear infinite}.nprogress-custom-parent{overflow:hidden;position:relative}.nprogress-custom-parent #nprogress .spinner,.nprogress-custom-parent #nprogress .bar{position:absolute}@-webkit-keyframes nprogress-spinner{0%{-webkit-transform:rotate(0deg)}to{-webkit-transform:rotate(360deg)}}@keyframes nprogress-spinner{0%{transform:rotate(0)}to{transform:rotate(360deg)}}*{scrollbar-color:var(--vp-c-divider-light) var(--vp-button-alt-bg);scrollbar-width:thin}::-webkit-scrollbar{width:8px;height:8px}::-webkit-scrollbar-thumb{background:var(--vp-c-mute)}::-webkit-scrollbar-thumb:hover{background:var(--vp-button-alt-hover-bg)}::-webkit-scrollbar-track{background:var(--vp-button-alt-bg)}::-webkit-scrollbar-corner{background:transparent}.katex{font-size:1.05em;direction:ltr}.katex-display{overflow:auto hidden;-webkit-overflow-scrolling:touch;padding-top:.2em;padding-bottom:.2em}.katex-display::-webkit-scrollbar{height:3px}.katex-display .katex{font-size:1.21em}.katex-error{color:red}.medium-zoom-overlay{z-index:20}.medium-zoom-image{z-index:21}.medium-zoom-image{align-items:center}.VPDoc .content-container img{border-radius:5px}.heart{animation:heart 5s ease infinite;color:red}@keyframes heart{0%{transform:scale(1)}5%{transform:scale(1.25)}20%{transform:scale(1)}30%{transform:scale(1)}35%{transform:scale(1.25)}50%{transform:scale(1)}55%{transform:scale(1.25)}70%{transform:scale(1)}}.coffee{color:#580d0d}pre,code,kbd,samp{font-family:var(--vp-font-family-mono)}img.VPImage.image-src{width:300px}.vp-doc>div{width:100%}@media (min-width: 1440px){.VPDoc.has-aside .content-container.content-container{max-width:100%}}.vp-doc p a+a{margin-left:17px}@media (max-width: 1080px){:root{--vp-sidebar-width: 260px}.VPNav .DocSearch-Button-Placeholder{display:none}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:10px;padding-right:10px}}@media (max-width: 992px){.VPNav .VPNavBar{padding-left:24px;padding-right:24px}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:8px;padding-right:8px}}@media (max-width: 800px){.VPNav .VPNavBarSearch{padding-left:16px}.VPNav .VPNavBarMenuLink,.VPNav .VPNavBarMenuGroup .button{padding-left:6px;padding-right:6px}}.VPMenuGroup .title{white-space:nowrap}.DocSearch-Logo{display:none}.link-preview-widget{display:table;width:100%;border:2px solid rgba(55,53,47,.16);border-radius:5px}.link-preview-widget-title{font-size:16px;font-weight:700;display:-webkit-box;overflow:hidden;margin-bottom:8px;word-break:break-all;-webkit-box-orient:vertical;-webkit-line-clamp:2}.link-preview-widget-description{font-size:12px;font-style:normal;line-height:1.5;display:-webkit-box;overflow:hidden;max-height:3em;margin-bottom:4px;word-break:break-all;color:#37352fa6;-webkit-box-orient:vertical;-webkit-line-clamp:2}.link-preview-widget-url{font-size:12px;font-style:normal;line-height:1.5;display:block;margin-bottom:0;color:#37352fa6}.link-preview-widget>a{text-decoration:none!important;display:table-cell;flex-direction:column;padding:16px;cursor:pointer;vertical-align:middle;color:inherit;background-color:transparent;-webkit-box-orient:vertical;-webkit-box-direction:normal;-webkit-box-flex:1;flex:1;transition:background-color .2s ease-in-out;border-radius:5px}.link-preview-widget>a:hover{background-color:var(--vp-custom-block-tip-bg)}.link-preview-widget-image{width:260px;min-width:220px;height:150px;padding:0;vertical-align:middle;border-left:1px rgba(55,53,47,.16) #f2f2f2;border-radius:5px;background-repeat:no-repeat;background-position:50%;background-size:cover;-webkit-box-flex:0;flex:0}html.dark .link-preview-widget{display:table;width:100%;border:2px solid rgba(200,202,208,.16);border-radius:5px}html.dark .link-preview-widget-title{font-size:16px;font-weight:700;display:-webkit-box;overflow:hidden;margin-bottom:8px;word-break:break-all;-webkit-box-orient:vertical;-webkit-line-clamp:2}html.dark .link-preview-widget-description{font-size:12px;font-style:normal;line-height:1.5;display:-webkit-box;overflow:hidden;max-height:3em;margin-bottom:4px;word-break:break-all;color:var(--vp-c-text-2);-webkit-box-orient:vertical;-webkit-line-clamp:2}html.dark .link-preview-widget-url{font-size:12px;font-style:normal;line-height:1.5;display:block;margin-bottom:0;color:var(--vp-c-text-2)}html.dark .link-preview-widget>a{text-decoration:none!important;display:table-cell;flex-direction:column;padding:16px;cursor:pointer;vertical-align:middle;color:inherit;background-color:transparent;-webkit-box-orient:vertical;-webkit-box-direction:normal;-webkit-box-flex:1;flex:1;transition:background-color .2s ease-in-out}html.dark .link-preview-widget>a:hover{background-color:var(--vp-custom-block-tip-bg)}html.dark .link-preview-widget-image{width:260px;min-width:220px;height:150px;padding:0;vertical-align:middle;border-left:1px rgba(200,202,208,.16) #f2f2f2;border-radius:5px;background-repeat:no-repeat;background-position:50%;background-size:cover;-webkit-box-flex:0;flex:0}:root{--vp-home-hero-name-color: transparent;--vp-home-hero-name-background: -webkit-linear-gradient(315deg, #42d392 25%, #647eff);--vp-home-hero-image-background-image: linear-gradient(-45deg, #42d392 50%, #47caff 50%);--vp-home-hero-image-filter: blur(40px)}@media (min-width: 640px){:root{--vp-home-hero-image-filter: blur(56px)}}@media (min-width: 960px){:root{--vp-home-hero-image-filter: blur(72px)}}.vp-sponsor-grid.medium .vp-sponsor-grid-image{max-height:90px}.copyright[data-v-ddda9334]{border-color:var(--vp-custom-block-tip-border);border-style:solid;border-width:1px;background-color:var(--vp-custom-block-tip-bg);border-radius:6px;color:var(--vp-c-text-2);font-size:15px;margin-top:50px}.copyright .content[data-v-ddda9334]{padding:13px 16px}.copyright .content .item[data-v-ddda9334]{margin-bottom:5px;word-break:break-word;line-height:22px}.copyright .content .item .icon[data-v-ddda9334]{display:inline-block;height:16px;width:16px;margin-right:.375rem;vertical-align:-2.5px}a[data-v-ddda9334]{font-weight:400;color:var(--vp-c-text-2)}a[data-v-ddda9334]:hover{color:var(--vp-c-brand)}.copyrightText[data-v-ddda9334]{font-weight:700}/*! normalize.css v8.0.1 | MIT License | github.com/necolas/normalize.css */html{line-height:1.15;-webkit-text-size-adjust:100%;-webkit-font-smoothing:antialiased;-moz-osx-font-smoothing:grayscale}body{margin:0}main{display:block}h1{margin:.67em 0;font-size:2em}hr{box-sizing:content-box;height:0;overflow:visible}pre{font-size:1em;font-family:monospace,monospace}a{background-color:transparent}abbr[title]{text-decoration:underline;text-decoration:underline dotted;border-bottom:none}b,strong{font-weight:bolder}code,kbd,samp{font-size:1em;font-family:monospace,monospace}small{font-size:80%}sub,sup{position:relative;font-size:75%;line-height:0;vertical-align:baseline}sub{bottom:-.25em}sup{top:-.5em}img{border-style:none}button,input,optgroup,select,textarea{margin:0;font-size:100%;font-family:inherit;line-height:1.15}button,input{overflow:visible}button,select{text-transform:none}button,[type=button],[type=reset],[type=submit]{-webkit-appearance:button}button::-moz-focus-inner,[type=button]::-moz-focus-inner,[type=reset]::-moz-focus-inner,[type=submit]::-moz-focus-inner{padding:0;border-style:none}button:-moz-focusring,[type=button]:-moz-focusring,[type=reset]:-moz-focusring,[type=submit]:-moz-focusring{outline:1px dotted ButtonText}fieldset{padding:.35em .75em .625em}legend{display:table;box-sizing:border-box;max-width:100%;padding:0;color:inherit;white-space:normal}progress{vertical-align:baseline}textarea{overflow:auto}[type=checkbox],[type=radio]{box-sizing:border-box;padding:0}[type=number]::-webkit-inner-spin-button,[type=number]::-webkit-outer-spin-button{height:auto}[type=search]{outline-offset:-2px;-webkit-appearance:textfield}[type=search]::-webkit-search-decoration{-webkit-appearance:none}::-webkit-file-upload-button{font:inherit;-webkit-appearance:button}details{display:block}summary{display:list-item}template{display:none}[hidden]{display:none}.arco-icon{display:inline-block;width:1em;height:1em;color:inherit;font-style:normal;vertical-align:-2px;outline:none;stroke:currentColor}.arco-icon-loading,.arco-icon-spin{animation:arco-loading-circle 1s infinite cubic-bezier(0,0,1,1)}@keyframes arco-loading-circle{0%{transform:rotate(0)}to{transform:rotate(360deg)}}.arco-icon-hover{position:relative;display:inline-block;cursor:pointer;line-height:12px}.arco-icon-hover .arco-icon{position:relative}.arco-icon-hover:before{position:absolute;display:block;box-sizing:border-box;background-color:transparent;border-radius:var(--border-radius-circle);transition:background-color .1s cubic-bezier(0,0,1,1);content:""}.arco-icon-hover:hover:before{background-color:var(--color-fill-2)}.arco-icon-hover.arco-icon-hover-disabled:before{opacity:0}.arco-icon-hover:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-mini{line-height:12px}.arco-icon-hover-size-mini:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-small{line-height:12px}.arco-icon-hover-size-small:before{top:50%;left:50%;width:20px;height:20px;transform:translate(-50%,-50%)}.arco-icon-hover-size-large{line-height:12px}.arco-icon-hover-size-large:before{top:50%;left:50%;width:24px;height:24px;transform:translate(-50%,-50%)}.arco-icon-hover-size-huge{line-height:12px}.arco-icon-hover-size-huge:before{top:50%;left:50%;width:24px;height:24px;transform:translate(-50%,-50%)}.fade-in-standard-enter-from,.fade-in-standard-appear-from{opacity:0}.fade-in-standard-enter-to,.fade-in-standard-appear-to{opacity:1}.fade-in-standard-enter-active,.fade-in-standard-appear-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1)}.fade-in-standard-leave-from{opacity:1}.fade-in-standard-leave-to{opacity:0}.fade-in-standard-leave-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1)}.fade-in-enter-from,.fade-in-appear-from{opacity:0}.fade-in-enter-to,.fade-in-appear-to{opacity:1}.fade-in-enter-active,.fade-in-appear-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.fade-in-leave-from{opacity:1}.fade-in-leave-to{opacity:0}.fade-in-leave-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.zoom-in-enter-from,.zoom-in-appear-from{transform:scale(.5);opacity:0}.zoom-in-enter-to,.zoom-in-appear-to{transform:scale(1);opacity:1}.zoom-in-enter-active,.zoom-in-appear-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1),transform .3s cubic-bezier(.34,.69,.1,1)}.zoom-in-leave-from{transform:scale(1);opacity:1}.zoom-in-leave-to{transform:scale(.5);opacity:0}.zoom-in-leave-active{transition:opacity .3s cubic-bezier(.34,.69,.1,1),transform .3s cubic-bezier(.34,.69,.1,1)}.zoom-in-fade-out-enter-from,.zoom-in-fade-out-appear-from{transform:scale(.5);opacity:0}.zoom-in-fade-out-enter-to,.zoom-in-fade-out-appear-to{transform:scale(1);opacity:1}.zoom-in-fade-out-enter-active,.zoom-in-fade-out-appear-active{transition:opacity .3s cubic-bezier(.3,1.3,.3,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-fade-out-leave-from{transform:scale(1);opacity:1}.zoom-in-fade-out-leave-to{transform:scale(.5);opacity:0}.zoom-in-fade-out-leave-active{transition:opacity .3s cubic-bezier(.3,1.3,.3,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-big-enter-from,.zoom-in-big-appear-from{transform:scale(.5);opacity:0}.zoom-in-big-enter-to,.zoom-in-big-appear-to{transform:scale(1);opacity:1}.zoom-in-big-enter-active,.zoom-in-big-appear-active{transition:opacity .2s cubic-bezier(0,0,1,1),transform .2s cubic-bezier(0,0,1,1)}.zoom-in-big-leave-from{transform:scale(1);opacity:1}.zoom-in-big-leave-to{transform:scale(.2);opacity:0}.zoom-in-big-leave-active{transition:opacity .2s cubic-bezier(0,0,1,1),transform .2s cubic-bezier(0,0,1,1)}.zoom-in-left-enter-from,.zoom-in-left-appear-from{transform:scale(.1);opacity:.1}.zoom-in-left-enter-to,.zoom-in-left-appear-to{transform:scale(1);opacity:1}.zoom-in-left-enter-active,.zoom-in-left-appear-active{transform-origin:0 50%;transition:opacity .3s cubic-bezier(0,0,1,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-left-leave-from{transform:scale(1);opacity:1}.zoom-in-left-leave-to{transform:scale(.1);opacity:.1}.zoom-in-left-leave-active{transform-origin:0 50%;transition:opacity .3s cubic-bezier(0,0,1,1),transform .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-top-enter-from,.zoom-in-top-appear-from{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-top-enter-to,.zoom-in-top-appear-to{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-top-enter-active,.zoom-in-top-appear-active{transform-origin:0 0;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-top-leave-from{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-top-leave-to{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-top-leave-active{transform-origin:0 0;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-bottom-enter-from,.zoom-in-bottom-appear-from{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-bottom-enter-to,.zoom-in-bottom-appear-to{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-bottom-enter-active,.zoom-in-bottom-appear-active{transform-origin:100% 100%;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.zoom-in-bottom-leave-from{transform:scaleY(1) translateZ(0);opacity:1}.zoom-in-bottom-leave-to{transform:scaleY(.8) translateZ(0);opacity:0}.zoom-in-bottom-leave-active{transform-origin:100% 100%;transition:transform .3s cubic-bezier(.3,1.3,.3,1),opacity .3s cubic-bezier(.3,1.3,.3,1)}.slide-dynamic-origin-enter-from,.slide-dynamic-origin-appear-from{transform:scaleY(.9);transform-origin:0 0;opacity:0}.slide-dynamic-origin-enter-to,.slide-dynamic-origin-appear-to{transform:scaleY(1);transform-origin:0 0;opacity:1}.slide-dynamic-origin-enter-active,.slide-dynamic-origin-appear-active{transition:transform .2s cubic-bezier(.34,.69,.1,1),opacity .2s cubic-bezier(.34,.69,.1,1)}.slide-dynamic-origin-leave-from{transform:scaleY(1);transform-origin:0 0;opacity:1}.slide-dynamic-origin-leave-to{transform:scaleY(.9);transform-origin:0 0;opacity:0}.slide-dynamic-origin-leave-active{transition:transform .2s cubic-bezier(.34,.69,.1,1),opacity .2s cubic-bezier(.34,.69,.1,1)}.slide-left-enter-from,.slide-left-appear-from{transform:translate(-100%)}.slide-left-enter-to,.slide-left-appear-to{transform:translate(0)}.slide-left-enter-active,.slide-left-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-left-leave-from{transform:translate(0)}.slide-left-leave-to{transform:translate(-100%)}.slide-left-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-right-enter-from,.slide-right-appear-from{transform:translate(100%)}.slide-right-enter-to,.slide-right-appear-to{transform:translate(0)}.slide-right-enter-active,.slide-right-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-right-leave-from{transform:translate(0)}.slide-right-leave-to{transform:translate(100%)}.slide-right-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-top-enter-from,.slide-top-appear-from{transform:translateY(-100%)}.slide-top-enter-to,.slide-top-appear-to{transform:translateY(0)}.slide-top-enter-active,.slide-top-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-top-leave-from{transform:translateY(0)}.slide-top-leave-to{transform:translateY(-100%)}.slide-top-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-bottom-enter-from,.slide-bottom-appear-from{transform:translateY(100%)}.slide-bottom-enter-to,.slide-bottom-appear-to{transform:translateY(0)}.slide-bottom-enter-active,.slide-bottom-appear-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}.slide-bottom-leave-from{transform:translateY(0)}.slide-bottom-leave-to{transform:translateY(100%)}.slide-bottom-leave-active{transition:transform .3s cubic-bezier(.34,.69,.1,1)}body{--red-1: 255,236,232;--red-2: 253,205,197;--red-3: 251,172,163;--red-4: 249,137,129;--red-5: 247,101,96;--red-6: 245,63,63;--red-7: 203,39,45;--red-8: 161,21,30;--red-9: 119,8,19;--red-10: 77,0,10;--orangered-1: 255,243,232;--orangered-2: 253,221,195;--orangered-3: 252,197,159;--orangered-4: 250,172,123;--orangered-5: 249,144,87;--orangered-6: 247,114,52;--orangered-7: 204,81,32;--orangered-8: 162,53,17;--orangered-9: 119,31,6;--orangered-10: 77,14,0;--orange-1: 255,247,232;--orange-2: 255,228,186;--orange-3: 255,207,139;--orange-4: 255,182,93;--orange-5: 255,154,46;--orange-6: 255,125,0;--orange-7: 210,95,0;--orange-8: 166,69,0;--orange-9: 121,46,0;--orange-10: 77,27,0;--gold-1: 255,252,232;--gold-2: 253,244,191;--gold-3: 252,233,150;--gold-4: 250,220,109;--gold-5: 249,204,69;--gold-6: 247,186,30;--gold-7: 204,146,19;--gold-8: 162,109,10;--gold-9: 119,75,4;--gold-10: 77,45,0;--yellow-1: 254,255,232;--yellow-2: 254,254,190;--yellow-3: 253,250,148;--yellow-4: 252,242,107;--yellow-5: 251,232,66;--yellow-6: 250,220,25;--yellow-7: 207,175,15;--yellow-8: 163,132,8;--yellow-9: 120,93,3;--yellow-10: 77,56,0;--lime-1: 252,255,232;--lime-2: 237,248,187;--lime-3: 220,241,144;--lime-4: 201,233,104;--lime-5: 181,226,65;--lime-6: 159,219,29;--lime-7: 126,183,18;--lime-8: 95,148,10;--lime-9: 67,112,4;--lime-10: 42,77,0;--green-1: 232,255,234;--green-2: 175,240,181;--green-3: 123,225,136;--green-4: 76,210,99;--green-5: 35,195,67;--green-6: 0,180,42;--green-7: 0,154,41;--green-8: 0,128,38;--green-9: 0,102,34;--green-10: 0,77,28;--cyan-1: 232,255,251;--cyan-2: 183,244,236;--cyan-3: 137,233,224;--cyan-4: 94,223,214;--cyan-5: 55,212,207;--cyan-6: 20,201,201;--cyan-7: 13,165,170;--cyan-8: 7,130,139;--cyan-9: 3,97,108;--cyan-10: 0,66,77;--blue-1: 232,247,255;--blue-2: 195,231,254;--blue-3: 159,212,253;--blue-4: 123,192,252;--blue-5: 87,169,251;--blue-6: 52,145,250;--blue-7: 32,108,207;--blue-8: 17,75,163;--blue-9: 6,48,120;--blue-10: 0,26,77;--arcoblue-1: 232,243,255;--arcoblue-2: 190,218,255;--arcoblue-3: 148,191,255;--arcoblue-4: 106,161,255;--arcoblue-5: 64,128,255;--arcoblue-6: 22,93,255;--arcoblue-7: 14,66,210;--arcoblue-8: 7,44,166;--arcoblue-9: 3,26,121;--arcoblue-10: 0,13,77;--purple-1: 245,232,255;--purple-2: 221,190,246;--purple-3: 195,150,237;--purple-4: 168,113,227;--purple-5: 141,78,218;--purple-6: 114,46,209;--purple-7: 85,29,176;--purple-8: 60,16,143;--purple-9: 39,6,110;--purple-10: 22,0,77;--pinkpurple-1: 255,232,251;--pinkpurple-2: 247,186,239;--pinkpurple-3: 240,142,230;--pinkpurple-4: 232,101,223;--pinkpurple-5: 225,62,219;--pinkpurple-6: 217,26,217;--pinkpurple-7: 176,16,182;--pinkpurple-8: 138,9,147;--pinkpurple-9: 101,3,112;--pinkpurple-10: 66,0,77;--magenta-1: 255,232,241;--magenta-2: 253,194,219;--magenta-3: 251,157,199;--magenta-4: 249,121,183;--magenta-5: 247,84,168;--magenta-6: 245,49,157;--magenta-7: 203,30,131;--magenta-8: 161,16,105;--magenta-9: 119,6,79;--magenta-10: 77,0,52;--gray-1: 247,248,250;--gray-2: 242,243,245;--gray-3: 229,230,235;--gray-4: 201,205,212;--gray-5: 169,174,184;--gray-6: 134,144,156;--gray-7: 107,119,133;--gray-8: 78,89,105;--gray-9: 39,46,59;--gray-10: 29,33,41;--success-1: var(--green-1);--success-2: var(--green-2);--success-3: var(--green-3);--success-4: var(--green-4);--success-5: var(--green-5);--success-6: var(--green-6);--success-7: var(--green-7);--success-8: var(--green-8);--success-9: var(--green-9);--success-10: var(--green-10);--primary-1: var(--arcoblue-1);--primary-2: var(--arcoblue-2);--primary-3: var(--arcoblue-3);--primary-4: var(--arcoblue-4);--primary-5: var(--arcoblue-5);--primary-6: var(--arcoblue-6);--primary-7: var(--arcoblue-7);--primary-8: var(--arcoblue-8);--primary-9: var(--arcoblue-9);--primary-10: var(--arcoblue-10);--danger-1: var(--red-1);--danger-2: var(--red-2);--danger-3: var(--red-3);--danger-4: var(--red-4);--danger-5: var(--red-5);--danger-6: var(--red-6);--danger-7: var(--red-7);--danger-8: var(--red-8);--danger-9: var(--red-9);--danger-10: var(--red-10);--warning-1: var(--orange-1);--warning-2: var(--orange-2);--warning-3: var(--orange-3);--warning-4: var(--orange-4);--warning-5: var(--orange-5);--warning-6: var(--orange-6);--warning-7: var(--orange-7);--warning-8: var(--orange-8);--warning-9: var(--orange-9);--warning-10: var(--orange-10);--link-1: var(--arcoblue-1);--link-2: var(--arcoblue-2);--link-3: var(--arcoblue-3);--link-4: var(--arcoblue-4);--link-5: var(--arcoblue-5);--link-6: var(--arcoblue-6);--link-7: var(--arcoblue-7);--link-8: var(--arcoblue-8);--link-9: var(--arcoblue-9);--link-10: var(--arcoblue-10)}body[arco-theme=dark]{--red-1: 77,0,10;--red-2: 119,6,17;--red-3: 161,22,31;--red-4: 203,46,52;--red-5: 245,78,78;--red-6: 247,105,101;--red-7: 249,141,134;--red-8: 251,176,167;--red-9: 253,209,202;--red-10: 255,240,236;--orangered-1: 77,14,0;--orangered-2: 119,30,5;--orangered-3: 162,55,20;--orangered-4: 204,87,41;--orangered-5: 247,126,69;--orangered-6: 249,146,90;--orangered-7: 250,173,125;--orangered-8: 252,198,161;--orangered-9: 253,222,197;--orangered-10: 255,244,235;--orange-1: 77,27,0;--orange-2: 121,48,4;--orange-3: 166,75,10;--orange-4: 210,105,19;--orange-5: 255,141,31;--orange-6: 255,150,38;--orange-7: 255,179,87;--orange-8: 255,205,135;--orange-9: 255,227,184;--orange-10: 255,247,232;--gold-1: 77,45,0;--gold-2: 119,75,4;--gold-3: 162,111,15;--gold-4: 204,150,31;--gold-5: 247,192,52;--gold-6: 249,204,68;--gold-7: 250,220,108;--gold-8: 252,233,149;--gold-9: 253,244,190;--gold-10: 255,252,232;--yellow-1: 77,56,0;--yellow-2: 120,94,7;--yellow-3: 163,134,20;--yellow-4: 207,179,37;--yellow-5: 250,225,60;--yellow-6: 251,233,75;--yellow-7: 252,243,116;--yellow-8: 253,250,157;--yellow-9: 254,254,198;--yellow-10: 254,255,240;--lime-1: 42,77,0;--lime-2: 68,112,6;--lime-3: 98,148,18;--lime-4: 132,183,35;--lime-5: 168,219,57;--lime-6: 184,226,75;--lime-7: 203,233,112;--lime-8: 222,241,152;--lime-9: 238,248,194;--lime-10: 253,255,238;--green-1: 0,77,28;--green-2: 4,102,37;--green-3: 10,128,45;--green-4: 18,154,55;--green-5: 29,180,64;--green-6: 39,195,70;--green-7: 80,210,102;--green-8: 126,225,139;--green-9: 178,240,183;--green-10: 235,255,236;--cyan-1: 0,66,77;--cyan-2: 6,97,108;--cyan-3: 17,131,139;--cyan-4: 31,166,170;--cyan-5: 48,201,201;--cyan-6: 63,212,207;--cyan-7: 102,223,215;--cyan-8: 144,233,225;--cyan-9: 190,244,237;--cyan-10: 240,255,252;--blue-1: 0,26,77;--blue-2: 5,47,120;--blue-3: 19,76,163;--blue-4: 41,113,207;--blue-5: 70,154,250;--blue-6: 90,170,251;--blue-7: 125,193,252;--blue-8: 161,213,253;--blue-9: 198,232,254;--blue-10: 234,248,255;--arcoblue-1: 0,13,77;--arcoblue-2: 4,27,121;--arcoblue-3: 14,50,166;--arcoblue-4: 29,77,210;--arcoblue-5: 48,111,255;--arcoblue-6: 60,126,255;--arcoblue-7: 104,159,255;--arcoblue-8: 147,190,255;--arcoblue-9: 190,218,255;--arcoblue-10: 234,244,255;--purple-1: 22,0,77;--purple-2: 39,6,110;--purple-3: 62,19,143;--purple-4: 90,37,176;--purple-5: 123,61,209;--purple-6: 142,81,218;--purple-7: 169,116,227;--purple-8: 197,154,237;--purple-9: 223,194,246;--purple-10: 247,237,255;--pinkpurple-1: 66,0,77;--pinkpurple-2: 101,3,112;--pinkpurple-3: 138,13,147;--pinkpurple-4: 176,27,182;--pinkpurple-5: 217,46,217;--pinkpurple-6: 225,61,219;--pinkpurple-7: 232,102,223;--pinkpurple-8: 240,146,230;--pinkpurple-9: 247,193,240;--pinkpurple-10: 255,242,253;--magenta-1: 77,0,52;--magenta-2: 119,8,80;--magenta-3: 161,23,108;--magenta-4: 203,43,136;--magenta-5: 245,69,166;--magenta-6: 247,86,169;--magenta-7: 249,122,184;--magenta-8: 251,158,200;--magenta-9: 253,195,219;--magenta-10: 255,232,241;--gray-1: 23,23,26;--gray-2: 46,46,48;--gray-3: 72,72,73;--gray-4: 95,95,96;--gray-5: 120,120,122;--gray-6: 146,146,147;--gray-7: 171,171,172;--gray-8: 197,197,197;--gray-9: 223,223,223;--gray-10: 246,246,246;--primary-1: var(--arcoblue-1);--primary-2: var(--arcoblue-2);--primary-3: var(--arcoblue-3);--primary-4: var(--arcoblue-4);--primary-5: var(--arcoblue-5);--primary-6: var(--arcoblue-6);--primary-7: var(--arcoblue-7);--primary-8: var(--arcoblue-8);--primary-9: var(--arcoblue-9);--primary-10: var(--arcoblue-10);--success-1: var(--green-1);--success-2: var(--green-2);--success-3: var(--green-3);--success-4: var(--green-4);--success-5: var(--green-5);--success-6: var(--green-6);--success-7: var(--green-7);--success-8: var(--green-8);--success-9: var(--green-9);--success-10: var(--green-10);--danger-1: var(--red-1);--danger-2: var(--red-2);--danger-3: var(--red-3);--danger-4: var(--red-4);--danger-5: var(--red-5);--danger-6: var(--red-6);--danger-7: var(--red-7);--danger-8: var(--red-8);--danger-9: var(--red-9);--danger-10: var(--red-10);--warning-1: var(--orange-1);--warning-2: var(--orange-2);--warning-3: var(--orange-3);--warning-4: var(--orange-4);--warning-5: var(--orange-5);--warning-6: var(--orange-6);--warning-7: var(--orange-7);--warning-8: var(--orange-8);--warning-9: var(--orange-9);--warning-10: var(--orange-10);--link-1: var(--arcoblue-1);--link-2: var(--arcoblue-2);--link-3: var(--arcoblue-3);--link-4: var(--arcoblue-4);--link-5: var(--arcoblue-5);--link-6: var(--arcoblue-6);--link-7: var(--arcoblue-7);--link-8: var(--arcoblue-8);--link-9: var(--arcoblue-9);--link-10: var(--arcoblue-10)}body{--color-white: #ffffff;--color-black: #000000;--color-border: rgb(var(--gray-3));--color-bg-popup: var(--color-bg-5);--color-bg-1: #fff;--color-bg-2: #fff;--color-bg-3: #fff;--color-bg-4: #fff;--color-bg-5: #fff;--color-bg-white: #fff;--color-neutral-1: rgb(var(--gray-1));--color-neutral-2: rgb(var(--gray-2));--color-neutral-3: rgb(var(--gray-3));--color-neutral-4: rgb(var(--gray-4));--color-neutral-5: rgb(var(--gray-5));--color-neutral-6: rgb(var(--gray-6));--color-neutral-7: rgb(var(--gray-7));--color-neutral-8: rgb(var(--gray-8));--color-neutral-9: rgb(var(--gray-9));--color-neutral-10: rgb(var(--gray-10));--color-text-1: var(--color-neutral-10);--color-text-2: var(--color-neutral-8);--color-text-3: var(--color-neutral-6);--color-text-4: var(--color-neutral-4);--color-border-1: var(--color-neutral-2);--color-border-2: var(--color-neutral-3);--color-border-3: var(--color-neutral-4);--color-border-4: var(--color-neutral-6);--color-fill-1: var(--color-neutral-1);--color-fill-2: var(--color-neutral-2);--color-fill-3: var(--color-neutral-3);--color-fill-4: var(--color-neutral-4);--color-primary-light-1: rgb(var(--primary-1));--color-primary-light-2: rgb(var(--primary-2));--color-primary-light-3: rgb(var(--primary-3));--color-primary-light-4: rgb(var(--primary-4));--color-link-light-1: rgb(var(--link-1));--color-link-light-2: rgb(var(--link-2));--color-link-light-3: rgb(var(--link-3));--color-link-light-4: rgb(var(--link-4));--color-secondary: var(--color-neutral-2);--color-secondary-hover: var(--color-neutral-3);--color-secondary-active: var(--color-neutral-4);--color-secondary-disabled: var(--color-neutral-1);--color-danger-light-1: rgb(var(--danger-1));--color-danger-light-2: rgb(var(--danger-2));--color-danger-light-3: rgb(var(--danger-3));--color-danger-light-4: rgb(var(--danger-4));--color-success-light-1: rgb(var(--success-1));--color-success-light-2: rgb(var(--success-2));--color-success-light-3: rgb(var(--success-3));--color-success-light-4: rgb(var(--success-4));--color-warning-light-1: rgb(var(--warning-1));--color-warning-light-2: rgb(var(--warning-2));--color-warning-light-3: rgb(var(--warning-3));--color-warning-light-4: rgb(var(--warning-4));--border-radius-none: 0;--border-radius-small: 2px;--border-radius-medium: 4px;--border-radius-large: 8px;--border-radius-circle: 50%;--color-tooltip-bg: rgb(var(--gray-10));--color-spin-layer-bg: rgba(255, 255, 255, .6);--color-menu-dark-bg: #232324;--color-menu-light-bg: #ffffff;--color-menu-dark-hover: rgba(255, 255, 255, .04);--color-mask-bg: rgba(29, 33, 41, .6)}body[arco-theme=dark]{--color-white: rgba(255, 255, 255, .9);--color-black: #000000;--color-border: #333335;--color-bg-1: #17171a;--color-bg-2: #232324;--color-bg-3: #2a2a2b;--color-bg-4: #313132;--color-bg-5: #373739;--color-bg-white: #f6f6f6;--color-text-1: rgba(255, 255, 255, .9);--color-text-2: rgba(255, 255, 255, .7);--color-text-3: rgba(255, 255, 255, .5);--color-text-4: rgba(255, 255, 255, .3);--color-fill-1: rgba(255, 255, 255, .04);--color-fill-2: rgba(255, 255, 255, .08);--color-fill-3: rgba(255, 255, 255, .12);--color-fill-4: rgba(255, 255, 255, .16);--color-primary-light-1: rgba(var(--primary-6), .2);--color-primary-light-2: rgba(var(--primary-6), .35);--color-primary-light-3: rgba(var(--primary-6), .5);--color-primary-light-4: rgba(var(--primary-6), .65);--color-secondary: rgba(var(--gray-9), .08);--color-secondary-hover: rgba(var(--gray-8), .16);--color-secondary-active: rgba(var(--gray-7), .24);--color-secondary-disabled: rgba(var(--gray-9), .08);--color-danger-light-1: rgba(var(--danger-6), .2);--color-danger-light-2: rgba(var(--danger-6), .35);--color-danger-light-3: rgba(var(--danger-6), .5);--color-danger-light-4: rgba(var(--danger-6), .65);--color-success-light-1: rgb(var(--success-6), .2);--color-success-light-2: rgb(var(--success-6), .35);--color-success-light-3: rgb(var(--success-6), .5);--color-success-light-4: rgb(var(--success-6), .65);--color-warning-light-1: rgb(var(--warning-6), .2);--color-warning-light-2: rgb(var(--warning-6), .35);--color-warning-light-3: rgb(var(--warning-6), .5);--color-warning-light-4: rgb(var(--warning-6), .65);--color-link-light-1: rgb(var(--link-6), .2);--color-link-light-2: rgb(var(--link-6), .35);--color-link-light-3: rgb(var(--link-6), .5);--color-link-light-4: rgb(var(--link-6), .65);--color-tooltip-bg: #373739;--color-spin-layer-bg: rgba(51, 51, 51, .6);--color-menu-dark-bg: #232324;--color-menu-light-bg: #232324;--color-menu-dark-hover: var(--color-fill-2);--color-mask-bg: rgba(23, 23, 26, .6)}body{font-size:14px;font-family:Inter,-apple-system,BlinkMacSystemFont,PingFang SC,Hiragino Sans GB,noto sans,Microsoft YaHei,Helvetica Neue,Helvetica,Arial,sans-serif}.arco-message-list{position:fixed;z-index:1003;display:flex;flex-direction:column;align-items:center;box-sizing:border-box;width:100%;margin:0;padding:0 10px;text-align:center;pointer-events:none}.arco-message-list-top{top:40px}.arco-message-list-bottom{bottom:40px}.arco-message{position:relative;display:inline-flex;align-items:center;margin-bottom:16px;padding:10px 16px;overflow:hidden;line-height:1;text-align:center;list-style:none;background-color:var(--color-bg-popup);border:1px solid var(--color-neutral-3);border-radius:var(--border-radius-small);box-shadow:0 4px 10px #0000001a;transition:all .1s cubic-bezier(0,0,1,1);pointer-events:auto}.arco-message-icon{display:inline-block;margin-right:8px;color:var(--color-text-1);font-size:20px;vertical-align:middle;animation:arco-msg-fade .1s cubic-bezier(0,0,1,1),arco-msg-fade .4s cubic-bezier(.3,1.3,.3,1)}.arco-message-content{font-size:14px;color:var(--color-text-1);vertical-align:middle}.arco-message-info{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-info .arco-message-icon{color:rgb(var(--primary-6))}.arco-message-info .arco-message-content{color:var(--color-text-1)}.arco-message-success{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-success .arco-message-icon{color:rgb(var(--success-6))}.arco-message-success .arco-message-content{color:var(--color-text-1)}.arco-message-warning{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-warning .arco-message-icon{color:rgb(var(--warning-6))}.arco-message-warning .arco-message-content{color:var(--color-text-1)}.arco-message-error{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-error .arco-message-icon{color:rgb(var(--danger-6))}.arco-message-error .arco-message-content{color:var(--color-text-1)}.arco-message-loading{background-color:var(--color-bg-popup);border-color:var(--color-neutral-3)}.arco-message-loading .arco-message-icon{color:rgb(var(--primary-6))}.arco-message-loading .arco-message-content{color:var(--color-text-1)}.arco-message-close-btn{margin-left:8px;color:var(--color-text-1);font-size:12px}.arco-message .arco-icon-hover.arco-message-icon-hover:before{width:20px;height:20px}.fade-message-enter-from,.fade-message-appear-from{opacity:0}.fade-message-enter-to,.fade-message-appear-to{opacity:1}.fade-message-enter-active,.fade-message-appear-active{transition:opacity .1s cubic-bezier(0,0,1,1)}.fade-message-leave-from{opacity:1}.fade-message-leave-to{opacity:0}.fade-message-leave-active{position:absolute}.flip-list-move{transition:transform .8s ease}@keyframes arco-msg-fade{0%{opacity:0}to{opacity:1}}@keyframes arco-msg-scale{0%{transform:scale(0)}to{transform:scale(1)}}:root{--c-brand: #3eaf7c;--c-brand-light: #4abf8a;--c-bg: #ffffff;--c-bg-light: #f3f4f5;--c-bg-lighter: #eeeeee;--c-bg-navbar: var(--c-bg);--c-bg-sidebar: var(--c-bg);--c-bg-arrow: #cccccc;--c-text: #2c3e50;--c-text-accent: var(--c-brand);--c-text-light: #3a5169;--c-text-lighter: #4e6e8e;--c-text-lightest: #6a8bad;--c-text-quote: #999999;--c-border: #eaecef;--c-border-dark: #dfe2e5;--c-tip: #42b983;--c-tip-bg: var(--c-bg-light);--c-tip-title: var(--c-text);--c-tip-text: var(--c-text);--c-tip-text-accent: var(--c-text-accent);--c-warning: #e7c000;--c-warning-bg: #fffae3;--c-warning-title: #ad9000;--c-warning-text: #746000;--c-warning-text-accent: var(--c-text);--c-danger: #cc0000;--c-danger-bg: #ffe0e0;--c-danger-title: #990000;--c-danger-text: #660000;--c-danger-text-accent: var(--c-text);--c-details-bg: #eeeeee;--c-badge-tip: var(--c-tip);--c-badge-warning: var(--c-warning);--c-badge-danger: var(--c-danger);--t-color: .3s ease;--t-transform: .3s ease;--code-bg-color: #282c34;--code-hl-bg-color: rgba(0, 0, 0, .66);--code-ln-color: #9e9e9e;--code-ln-wrapper-width: 3.5rem;--font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif;--font-family-code: Consolas, Monaco, "Andale Mono", "Ubuntu Mono", monospace;--navbar-height: 3.6rem;--navbar-padding-v: .7rem;--navbar-padding-h: 1.5rem;--sidebar-width: 20rem;--sidebar-width-mobile: calc(var(--sidebar-width) * .82);--content-width: 740px;--homepage-width: 960px}html.dark{--c-brand: #3aa675;--c-brand-light: #349469;--c-bg: #22272e;--c-bg-light: #2b313a;--c-bg-lighter: #262c34;--c-text: #adbac7;--c-text-light: #96a7b7;--c-text-lighter: #8b9eb0;--c-text-lightest: #8094a8;--c-border: #3e4c5a;--c-border-dark: #34404c;--c-tip: #318a62;--c-warning: #ceab00;--c-warning-bg: #7e755b;--c-warning-title: #ceac03;--c-warning-text: #362e00;--c-danger: #940000;--c-danger-bg: #806161;--c-danger-title: #610000;--c-danger-text: #3a0000;--c-details-bg: #323843;--code-hl-bg-color: #363b46}@font-face{font-family:octicons-link;src:url(data:font/woff;charset=utf-8;base64,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) format("woff")}.markdown-body{-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%;color:var(--c-text);font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Helvetica,Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji",Segoe UI Symbol;font-size:14px;line-height:1.5;word-wrap:break-word;transition:background-color var(--t-color),border-color var(--t-color)}.g-emoji{display:inline-block;min-width:1ch;font-family:"Apple Color Emoji","Segoe UI Emoji",Segoe UI Symbol,serif;font-size:1em;font-style:normal!important;font-weight:400;line-height:1;vertical-align:-.075em}.markdown-body .pl-c{color:#6a737d}.markdown-body .pl-c1,.markdown-body .pl-s .pl-v{color:#005cc5}.markdown-body .pl-e,.markdown-body .pl-en{color:#6f42c1}.markdown-body .pl-smi,.markdown-body .pl-s .pl-s1{color:var(--c-text)}.markdown-body .pl-ent{color:#22863a}.markdown-body .pl-k{color:#d73a49}.markdown-body .pl-s,.markdown-body .pl-pds,.markdown-body .pl-s .pl-pse .pl-s1,.markdown-body .pl-sr,.markdown-body .pl-sr .pl-cce,.markdown-body .pl-sr .pl-sre,.markdown-body .pl-sr .pl-sra{color:var(--c-text-lighter)}.markdown-body .pl-v,.markdown-body .pl-smw{color:#e36209}.markdown-body .pl-bu{color:#b31d28}.markdown-body .pl-ii{color:#fafbfc;background-color:#b31d28}.markdown-body .pl-c2{color:#fafbfc;background-color:#d73a49}.markdown-body .pl-c2:before{content:"^M"}.markdown-body .pl-sr .pl-cce{font-weight:700;color:#22863a}.markdown-body .pl-ml{color:#735c0f}.markdown-body .pl-mh,.markdown-body .pl-mh .pl-en,.markdown-body .pl-ms{font-weight:700;color:#005cc5}.markdown-body .pl-mi{font-style:italic;color:#24292e}.markdown-body .pl-mb{font-weight:700;color:#24292e}.markdown-body .pl-md{color:#b31d28;background-color:#ffeef0}.markdown-body .pl-mi1{color:#22863a;background-color:#f0fff4}.markdown-body .pl-mc{color:#e36209;background-color:#ffebda}.markdown-body .pl-mi2{color:#f6f8fa;background-color:#005cc5}.markdown-body .pl-mdr{font-weight:700;color:#6f42c1}.markdown-body .pl-ba{color:#586069}.markdown-body .pl-sg{color:#959da5}.markdown-body .pl-corl{text-decoration:underline;color:var(--c-text-lighter)}.markdown-body .octicon{display:inline-block;vertical-align:text-top;fill:currentColor}.markdown-body a{background-color:transparent;-webkit-text-decoration-skip:objects}.markdown-body a:active,.markdown-body a:hover{outline-width:0}.markdown-body strong{font-weight:inherit}.markdown-body strong{font-weight:bolder}.markdown-body h1{font-size:2em;margin:.67em 0}.markdown-body img{border-style:none}.markdown-body svg:not(:root){overflow:hidden}.markdown-body code,.markdown-body kbd,.markdown-body pre{font-family:monospace,monospace;font-size:1em}.markdown-body hr{-webkit-box-sizing:content-box;box-sizing:content-box;height:0;overflow:visible}.markdown-body input{font:inherit;margin:0}.markdown-body input{overflow:visible}.markdown-body [type=checkbox]{-webkit-box-sizing:border-box;box-sizing:border-box;padding:0}.markdown-body *{-webkit-box-sizing:border-box;box-sizing:border-box}.markdown-body input{font-family:inherit;font-size:inherit;line-height:inherit}.markdown-body a{color:#0366d6;text-decoration:none}.markdown-body a:hover{text-decoration:underline}.markdown-body strong{font-weight:600}.markdown-body hr{height:0;margin:15px 0;overflow:hidden;background:transparent;border:0;border-bottom:1px solid #dfe2e5}.markdown-body hr:before{display:table;content:""}.markdown-body hr:after{display:table;clear:both;content:""}.markdown-body table{border-spacing:0;border-collapse:collapse}.markdown-body td,.markdown-body th{padding:0}.markdown-body h1,.markdown-body h2,.markdown-body h3,.markdown-body h4,.markdown-body h5,.markdown-body h6{margin-top:0;margin-bottom:0}.markdown-body h1{font-size:32px;font-weight:600}.markdown-body h2{font-size:24px;font-weight:600}.markdown-body h3{font-size:20px;font-weight:600}.markdown-body h4,.markdown-body h5{font-size:14px;font-weight:600}.markdown-body h6{font-size:12px;font-weight:600}.markdown-body p{margin-top:0;margin-bottom:10px}.markdown-body blockquote{margin:0}.markdown-body ul,.markdown-body ol{padding-left:0;margin-top:0;margin-bottom:0}.markdown-body ol ol,.markdown-body ul ol{list-style-type:lower-roman}.markdown-body ul ul ol,.markdown-body ul ol ol,.markdown-body ol ul ol,.markdown-body ol ol ol{list-style-type:lower-alpha}.markdown-body dd{margin-left:0}.markdown-body code{font-family:SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace;font-size:12px}.markdown-body pre{margin-top:0;margin-bottom:0;font:12px SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace}.markdown-body .octicon{vertical-align:text-bottom}.markdown-body .pl-0{padding-left:0!important}.markdown-body .pl-1{padding-left:4px!important}.markdown-body .pl-2{padding-left:8px!important}.markdown-body .pl-3{padding-left:14px!important}.markdown-body .pl-4{padding-left:24px!important}.markdown-body .pl-5{padding-left:32px!important}.markdown-body .pl-6{padding-left:40px!important}.markdown-body:before{display:table;content:""}.markdown-body:after{display:table;clear:both;content:""}.markdown-body>*:first-child{margin-top:0!important}.markdown-body>*:last-child{margin-bottom:0!important}.markdown-body a:not([href]){color:inherit;text-decoration:none}.markdown-body .anchor{float:left;padding-right:4px;margin-left:-20px;line-height:1}.markdown-body .anchor:focus{outline:none}.markdown-body p,.markdown-body blockquote,.markdown-body ul,.markdown-body ol,.markdown-body dl,.markdown-body table,.markdown-body pre{margin-top:0;margin-bottom:14px}.markdown-body hr{height:.25em;padding:0;margin:24px 0;background-color:#e1e4e8;border:0}.markdown-body blockquote{padding:0 1em;color:#6a737d;border-left:.25em solid var(--c-border)}.markdown-body blockquote>:first-child{margin-top:0}.markdown-body blockquote>:last-child{margin-bottom:0}.markdown-body kbd{display:inline-block;padding:3px 5px;font-size:11px;line-height:10px;color:#444d56;vertical-align:middle;background-color:#fafbfc;border:solid 1px #c6cbd1;border-bottom-color:#959da5;border-radius:3px;-webkit-box-shadow:inset 0 -1px 0 #959da5;box-shadow:inset 0 -1px #959da5}.markdown-body h1,.markdown-body h2,.markdown-body h3,.markdown-body h4,.markdown-body h5,.markdown-body h6{margin-top:24px;margin-bottom:14px;font-weight:600;line-height:1.25}.markdown-body h1 .octicon-link,.markdown-body h2 .octicon-link,.markdown-body h3 .octicon-link,.markdown-body h4 .octicon-link,.markdown-body h5 .octicon-link,.markdown-body h6 .octicon-link{color:#1b1f23;vertical-align:middle;visibility:hidden}.markdown-body h1:hover .anchor,.markdown-body h2:hover .anchor,.markdown-body h3:hover .anchor,.markdown-body h4:hover .anchor,.markdown-body h5:hover .anchor,.markdown-body h6:hover .anchor{text-decoration:none}.markdown-body h1:hover .anchor .octicon-link,.markdown-body h2:hover .anchor .octicon-link,.markdown-body h3:hover .anchor .octicon-link,.markdown-body h4:hover .anchor .octicon-link,.markdown-body h5:hover .anchor .octicon-link,.markdown-body h6:hover .anchor .octicon-link{visibility:visible}.markdown-body h1{padding-bottom:.3em;font-size:2em;border-bottom:1px solid #eaecef}.markdown-body h2{padding-bottom:.3em;font-size:1.5em;border-bottom:1px solid #eaecef}.markdown-body h3{font-size:1.25em}.markdown-body h4{font-size:1em}.markdown-body h5{font-size:.875em}.markdown-body h6{font-size:.85em;color:#6a737d}.markdown-body ul,.markdown-body ol{padding-left:2em}.markdown-body ul ul,.markdown-body ul ol,.markdown-body ol ol,.markdown-body ol ul{margin-top:0;margin-bottom:0}.markdown-body li>p{margin-top:14px}.markdown-body li+li{margin-top:.25em}.markdown-body dl{padding:0}.markdown-body dl dt{padding:0;margin-top:14px;font-size:1em;font-style:italic;font-weight:600}.markdown-body dl dd{padding:0 14px;margin-bottom:14px}.markdown-body table{display:block;width:100%;overflow:auto}.markdown-body table th{font-weight:600}.markdown-body table th,.markdown-body table td{padding:6px 13px;border:1px solid #dfe2e5}.markdown-body table tr{background-color:#fff;border-top:1px solid #c6cbd1}.markdown-body table tr:nth-child(2n){background-color:#f6f8fa}.markdown-body img{max-width:100%;-webkit-box-sizing:content-box;box-sizing:content-box;background-color:#fff}.markdown-body code{padding:.2em 0;margin:0;font-size:85%;border-radius:3px}.markdown-body code:before,.markdown-body code:after{letter-spacing:-.2em;content:" "}.markdown-body pre{word-wrap:normal}.markdown-body pre>code{padding:0;margin:0;font-size:100%;word-break:normal;white-space:pre;background:transparent;border:0}.markdown-body .highlight{margin-bottom:14px}.markdown-body .highlight pre{margin-bottom:0;word-break:normal}.markdown-body .highlight pre,.markdown-body pre{padding:14px;overflow:auto;font-size:85%;line-height:1.45;background-color:var(--c-tip-bg);border-radius:3px;color:var(--c-text);transition:background-color var(--t-color),border-color var(--t-color)}.markdown-body pre code{display:inline;max-width:auto;padding:0;margin:0;overflow:visible;line-height:inherit;word-wrap:normal;background-color:transparent;border:0;color:var(--c-text)}.markdown-body pre code:before,.markdown-body pre code:after{content:normal}.markdown-body .full-commit .btn-outline:not(:disabled):hover{color:#005cc5;border-color:#005cc5}.markdown-body kbd{display:inline-block;padding:3px 5px;font:11px SFMono-Regular,Consolas,Liberation Mono,Menlo,Courier,monospace;line-height:10px;color:#444d56;vertical-align:middle;background-color:#fafbfc;border:solid 1px #d1d5da;border-bottom-color:#c6cbd1;border-radius:3px;-webkit-box-shadow:inset 0 -1px 0 #c6cbd1;box-shadow:inset 0 -1px #c6cbd1}.markdown-body :checked+.radio-label{position:relative;z-index:1;border-color:#0366d6}.markdown-body .task-list-item{list-style-type:none}.markdown-body .task-list-item+.task-list-item{margin-top:3px}.markdown-body .task-list-item input{margin:0 .2em .25em -1.6em;vertical-align:middle}.markdown-body hr{border-bottom-color:#eee}.gt-container{-webkit-box-sizing:border-box;box-sizing:border-box;font-size:14px}.gt-container *{-webkit-box-sizing:border-box;box-sizing:border-box}.gt-container a{color:var(--c-text-accent);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container a.is--active{color:var(--c-text);cursor:default!important}.gt-container a.is--active:hover{color:var(--c-text-accent)}.gt-container .hide{display:none!important}.gt-container .gt-svg{display:inline-block;width:1em;height:1em;vertical-align:sub}.gt-container .gt-ico-arrdown{fill:var(--c-text)}.gt-container .gt-svg svg{width:100%;height:100%;fill:var(--c-text-accent)}.gt-container .gt-ico{display:inline-block}.gt-container .gt-ico-text{margin-left:.3125em}.gt-container .gt-ico-tip{font-size:.875em}.gt-container .gt-ico-github,.gt-container .gt-ico-github .gt-svg{width:100%;height:100%}.gt-container .gt-ico-github svg{fill:var(--c-text)}.gt-container .gt-spinner{position:relative}.gt-container .gt-spinner:before{content:"";-webkit-box-sizing:border-box;box-sizing:border-box;position:absolute;top:3px;width:.75em;height:.75em;margin-top:-.1875em;margin-left:-.375em;border-radius:50%;border:1px solid #fff;border-top-color:var(--c-text-accent);-webkit-animation:gt-kf-rotate .6s linear infinite;animation:gt-kf-rotate .6s linear infinite}.gt-container .gt-loader{position:relative;border:1px solid #999;-webkit-animation:ease gt-kf-rotate 1.5s infinite;animation:ease gt-kf-rotate 1.5s infinite;display:inline-block;font-style:normal;width:1.75em;height:1.75em;line-height:1.75em;border-radius:50%}.gt-container .gt-loader:before{content:"";position:absolute;display:block;top:0;left:50%;margin-top:-.1875em;margin-left:-.1875em;width:.375em;height:.375em;background-color:#999;border-radius:50%}.gt-container .gt-avatar{display:inline-block;width:3.125em;height:3.125em}@media (max-width: 479px){.gt-container .gt-avatar{width:2em;height:2em}}.gt-container .gt-avatar img{width:100%;height:auto;border-radius:3px}.gt-container .gt-avatar-github{width:3.125em;height:3.125em;cursor:pointer}@media (max-width: 479px){.gt-container .gt-avatar-github{width:2em;height:2em}}.gt-container .gt-btn{padding:.75em .95em;display:inline-block;line-height:1;text-decoration:none;white-space:nowrap;cursor:pointer;border:1px solid var(--c-text-accent);border-radius:5px;background-color:var(--c-brand);color:#fff;outline:none;font-size:.75em;transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-btn-text{font-weight:400}.gt-header-controls .gt-btn-public .gt-btn-text{color:var(--c-bg)}.gt-container .gt-btn-loading{position:relative;margin-left:.5em;display:inline-block;width:.75em;height:1em;vertical-align:top}.gt-container .gt-btn.is--disable{cursor:not-allowed;opacity:.5}.gt-container .gt-btn-login{margin-right:0;color:var(--c-bg)}.gt-container .gt-btn-login:hover{background-color:var(--c-brand-light);border-color:var(--c-brand)}.gt-container .gt-btn-preview{background-color:var(--c-bg);color:var(--c-brand);border-color:var(--c-brand);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-btn-preview:hover{color:var(--c-bg);background-color:var(--c-brand-light)}.gt-container .gt-btn-public:hover,.gt-container .gt-btn-loadmore:hover{background-color:var(--c-brand-light);border-color:var(--c-brand)}.gt-container .gt-btn-loadmore{color:var(--c-bg)}.gt-container .gt-error{text-align:center;margin:.625em;color:#ff3860}.gt-container .gt-initing{padding:1.25em 0;text-align:center}.gt-container .gt-initing-text{margin:.625em auto;font-size:92%}.gt-container .gt-no-init{padding:1.25em 0;text-align:center}.gt-container .gt-link{border-bottom:1px dotted var(--c-text-accent)}.gt-container .gt-link-counts,.gt-container .gt-link-project{text-decoration:none}.gt-container .gt-meta{margin:1.25em 0;padding:1em 0;position:relative;border-bottom:1px solid var(--c-border);font-size:1em;transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-meta:before,.gt-container .gt-meta:after{content:" ";display:table}.gt-container .gt-meta:after{clear:both}.gt-container .gt-counts{margin:0 .625em 0 0}.gt-container .gt-user{float:right;margin:0;font-size:92%}.gt-container .gt-user-pic{width:14px;height:14px;vertical-align:top;margin-right:.5em}.gt-container .gt-user-inner{display:inline-block;cursor:pointer}.gt-container .gt-user .gt-ico{margin:0 0 0 .3125em}.gt-container .gt-user .gt-ico svg{fill:inherit}.gt-container .gt-user .is--poping .gt-ico svg{fill:var(--c-text-accent)}.gt-container .gt-version{color:#a1a1a1;margin-left:.375em}.gt-container .gt-copyright{margin:0 .9375em .5em;border-top:1px solid var(--c-border);padding-top:.5em}.gt-container .gt-popup{position:absolute;right:0;top:2.375em;background-color:var(--c-bg-navbar);display:block!important;border:1px solid var(--c-border);padding:.625em 0;font-size:.875em;letter-spacing:.5px;box-sizing:border-box;z-index:10}.gt-container .gt-popup .gt-action{cursor:pointer;display:block;margin:.5em 0;padding:0 1.125em;position:relative;text-decoration:none}.gt-container .gt-popup .gt-action.is--active:before{content:"";width:.25em;height:.25em;background:var(--c-text-accent);position:absolute;left:.5em;top:.4375em}.gt-container .gt-header{position:relative;display:-webkit-box;display:-ms-flexbox;display:flex}.gt-container .gt-header-comment{-webkit-box-flex:1;-ms-flex:1;flex:1;padding-left:1.25em;width:90%}@media (max-width: 479px){.gt-container .gt-header-comment{padding-left:.875em}}.gt-container .gt-header-textarea{padding:.75em;display:block;-webkit-box-sizing:border-box;box-sizing:border-box;width:100%;min-height:5.125em;max-height:15em;border-radius:5px;border:1px solid var(--c-border);font-size:.875em;word-wrap:break-word;resize:vertical;background-color:var(--c-bg);outline:none;-webkit-transition:all .25s ease;transition:background-color var(--t-color),border-color var(--t-color)}.gt-header-textarea{color:var(--c-text)}.gt-container .gt-header-preview{padding:.75em;border-radius:5px;border:1px solid var(--c-border);background-color:var(--c-bg)}.gt-container .gt-header-controls{position:relative;margin:.75em 0 0}.gt-container .gt-header-controls:before,.gt-container .gt-header-controls:after{content:" ";display:table}.gt-container .gt-header-controls:after{clear:both}@media (max-width: 479px){.gt-container .gt-header-controls{margin:0}}.gt-container .gt-header-controls-tip{font-size:13px;color:var(--c-text-accent);text-decoration:none;vertical-align:sub}@media (max-width: 479px){.gt-container .gt-header-controls-tip{display:none}}.gt-container .gt-header-controls .gt-btn{float:right;margin-left:1.2em}@media (max-width: 479px){.gt-container .gt-header-controls .gt-btn{float:none;width:100%;margin:.75em 0 0}}.gt-container:after{content:"";position:fixed;bottom:100%;left:0;right:0;top:0;opacity:0}.gt-container.gt-input-focused{position:relative}.gt-container.gt-input-focused:after{content:"";position:fixed;bottom:0%;left:0;right:0;top:0;background:#000;opacity:.6;-webkit-transition:opacity .3s,bottom 0s;transition:opacity .3s,bottom 0s;z-index:9999}.gt-container.gt-input-focused .gt-header-comment{z-index:10000}.gt-container .gt-comments{padding-top:1.25em}.gt-container .gt-comments-null{text-align:center}.gt-container .gt-comments-controls{margin:1.25em 0;text-align:center}.gt-container .gt-comment{position:relative;padding:.625em 0;display:-webkit-box;display:-ms-flexbox;display:flex}.gt-container .gt-comment-content{-webkit-box-flex:1;-ms-flex:1;flex:1;margin-left:1.25em;padding:.75em 1em;background-color:var(--c-bg);overflow:auto;-webkit-transition:all ease .25s;border:1px solid var(--c-border);transition:background-color var(--t-color),border-color var(--t-color)}.gt-container .gt-comment-content{border-radius:5px}@media (max-width: 479px){.gt-container .gt-comment-content{margin-left:.875em;padding:.625em .75em}}.gt-container .gt-comment-header{margin-bottom:.75em;font-size:.875em;position:relative}.gt-container .gt-comment-block-1{float:right;height:1.375em;width:3.5em}.gt-container .gt-comment-block-2{float:right;height:1.375em;width:4em}.gt-container .gt-comment-username{font-weight:500;color:var(--c-text-accent);text-decoration:none}.gt-container .gt-comment-username:hover{text-decoration:underline}.gt-container .gt-comment-text,.gt-container .gt-comment-date{margin-left:.5em;color:#a1a1a1;font-size:.875em}.gt-container .gt-comment-like,.gt-container .gt-comment-edit,.gt-container .gt-comment-reply{position:absolute;height:1.375em}.gt-container .gt-comment-like:hover,.gt-container .gt-comment-edit:hover,.gt-container .gt-comment-reply:hover{cursor:pointer}.gt-container .gt-comment-like{top:0;right:2em}.gt-container .gt-comment-edit,.gt-container .gt-comment-reply{top:0;right:0}.gt-container .gt-comment-body{color:var(--c-text)}.gt-container .gt-comment-body .email-hidden-toggle a{display:inline-block;height:12px;padding:0 9px;font-size:12px;font-weight:600;line-height:6px;color:#444d56;text-decoration:none;vertical-align:middle;background:#dfe2e5;border-radius:1px}.gt-container .gt-comment-body .email-hidden-toggle a:hover{background-color:#c6cbd1}.gt-container .gt-comment-body .email-hidden-reply{display:none;white-space:pre-wrap}.gt-container .gt-comment-body .email-hidden-reply .email-signature-reply{padding:0 15px;margin:15px 0;color:#586069;border-left:4px solid #dfe2e5}.gt-container .gt-comment-body .email-hidden-reply.expanded{display:block}.gt-container .gt-comment-admin .gt-comment-content{background-color:var(--c-bg);border:1px solid var(--c-border)}@-webkit-keyframes gt-kf-rotate{0%{-webkit-transform:rotate(0);transform:rotate(0)}to{-webkit-transform:rotate(360deg);transform:rotate(360deg)}}@keyframes gt-kf-rotate{0%{-webkit-transform:rotate(0);transform:rotate(0)}to{-webkit-transform:rotate(360deg);transform:rotate(360deg)}}.gt-comment-body .snippet-clipboard-content pre code{color:var(--c-text)}
diff --git a/development/aws/README.html b/development/aws/README.html
index f7381383..28bff8fe 100644
--- a/development/aws/README.html
+++ b/development/aws/README.html
@@ -5,12 +5,12 @@
     
     Toshiki's Note | Toshiki's Note
     
-    
+    
     
     
   
   
-  
+  
   
   
   
@@ -32,9 +32,9 @@
   
   
   
-    
    Toshiki's Note
    Appearance
    TOC
    The source of the main text is here, written with asciidoctor.
    If you want to build the document from the source, follow these steps.
    Install ​
    Prerequisites:
    • ruby
    • bundler
    Install ruby libraries:
    bash
    bundle install --path vendor/bundle
    bundle install --path vendor/bundle
    Build the document ​
    bash
    ./generate.sh main.adoc
    ./generate.sh main.adoc
    This will generate HTML and PDF documents in the directory named dist/.
    Source structure ​
    • main.adoc: Main document
    • dist/: Output document (both HTML and PDF) will be generated here
    • imgs/: Put image files here
    • fonts/, pdf-style/: Fonts and styles for PDF generation. Do not touch!
    • XXX.adoc: One section per file
    
-    
+    
    Toshiki's Note
    Appearance
    TOC
    The source of the main text is here, written with asciidoctor.
    If you want to build the document from the source, follow these steps.
    Install ​
    Prerequisites:
    • ruby
    • bundler
    Install ruby libraries:
    bash
    bundle install --path vendor/bundle
    bundle install --path vendor/bundle
    Build the document ​
    bash
    ./generate.sh main.adoc
    ./generate.sh main.adoc
    This will generate HTML and PDF documents in the directory named dist/.
    Source structure ​
    • main.adoc: Main document
    • dist/: Output document (both HTML and PDF) will be generated here
    • imgs/: Put image files here
    • fonts/, pdf-style/: Fonts and styles for PDF generation. Do not touch!
    • XXX.adoc: One section per file
    
+    
     
   
 
\ No newline at end of file
diff --git a/development/aws/acknowledgement.html b/development/aws/acknowledgement.html
index ef2f811c..bcd1efbd 100644
--- a/development/aws/acknowledgement.html
+++ b/development/aws/acknowledgement.html
@@ -5,12 +5,12 @@
     
     謝辞 | Toshiki's Note
     
-    
+    
     
     
   
   
-  
+  
   
   
   
@@ -32,9 +32,9 @@
   
   
   
-    
    Toshiki's Note
    Appearance
    TOC
    謝辞 ​
    本原稿の執筆にあたり，以下の方々からの協力を得た．この場を借りて，感謝を表したい．
    2021 年バージョンの contributors
    • 香取真知子氏 - ハンズオンプログラムの動作確認，文章校閲
    2020 年バージョンの contributors
    • 勝俣敬寛氏 - Docker イメージの作成
    • 香取真知子氏 - ハンズオンプログラムの動作確認
    • @shuuji3 - MR !15
    • @takatama_jp - MR !14
    本書の執筆には Asciidoctor を使用した．
    また，本書はオープンソースの教科書として，すべての読者・ディベロッパーからのフィードバックを受け付けている． 誤植や記述の誤り，改善点など見つかったら，ぜひ Issues や Pull request を投稿していただきたい．
    
-    
+    
    Toshiki's Note
    Appearance
    TOC
    謝辞 ​
    本原稿の執筆にあたり，以下の方々からの協力を得た．この場を借りて，感謝を表したい．
    2021 年バージョンの contributors
    • 香取真知子氏 - ハンズオンプログラムの動作確認，文章校閲
    2020 年バージョンの contributors
    • 勝俣敬寛氏 - Docker イメージの作成
    • 香取真知子氏 - ハンズオンプログラムの動作確認
    • @shuuji3 - MR !15
    • @takatama_jp - MR !14
    本書の執筆には Asciidoctor を使用した．
    また，本書はオープンソースの教科書として，すべての読者・ディベロッパーからのフィードバックを受け付けている． 誤植や記述の誤り，改善点など見つかったら，ぜひ Issues や Pull request を投稿していただきたい．
    
+    
     
   
 
\ No newline at end of file
diff --git a/development/aws/appendix.html b/development/aws/appendix.html
index c4fd8059..198f6259 100644
--- a/development/aws/appendix.html
+++ b/development/aws/appendix.html
@@ -5,13 +5,13 @@
     
     Appendix: 環境構築 | Toshiki's Note
     
-    
+    
     
     
   
   
   
-  
+  
   
   
   
@@ -63,9 +63,9 @@
 export AWS_SECRET_ACCESS_KEY=YYYYYY
 export AWS_DEFAULT_REGION=ap-northeast-1
    これらの環境変数は， ~/.aws/credentials よりも高い優先度をもつので，環境変数が設定されていればそちらの情報が使用される (参照)．
    AWS Educate Starter Account は us-east-1 のリージョンのみ利用可能である (執筆時点での情報)． よって， AWS Educate Starter Account を使用している場合は， default region を us-east-1 に設定する必要がある．
    AWS CDK のインストール ​
    読者のために，執筆時点におけるインストールの手順 (Linux 向け) を簡単に記述する． 将来のバージョンでは変更される可能性があるので，常に 公式のドキュメンテーション で最新の情報をチェックすることを忘れずに．
    Node.js がインストールされていれば，基本的に次のコマンドを実行すればよい．
    shell
    $ sudo npm install -g aws-cdk
    $ sudo npm install -g aws-cdk
    本書のハンズオンは AWS CDK version 1.100.0 で開発した． CDK は開発途上のライブラリなので，将来的に API が変更される可能性がある． API の変更によりエラーが生じた場合は， version 1.100.0 を使用することを推奨する．
    shell
    $ npm install -g aws-cdk@1.100
    $ npm install -g aws-cdk@1.100
    インストールできたか確認するため，次のコマンドを打って正しくバージョンが表示されることを確認する．
    shell
    $ cdk --version
    $ cdk --version
    インストールができたら，次のコマンドにより AWS 側の初期設定を行う． これは一度実行すれば OK．
    shell
    $ cdk bootstrap
    $ cdk bootstrap
    cdk bootstrap を実行するときは，AWS の認証情報とリージョンが正しく設定されていることを確認する． デフォルトでは ~/.aws/config にあるデフォルトのプロファイルが使用される． デフォルト以外のプロファイルを用いるときは AWS CLI のインストール で紹介したテクニックを使って切り替える．
    AWS CDK の認証情報の設定は AWS CLI と基本的に同じである．詳しくは AWS CLI のインストール を参照．
    WSL のインストール ​
    本書のハンズオンではコマンドラインから AWS CLI のコマンドを実行したり， Python で書かれたプログラムを実行する． コマンドは基本的に UNIX のターミナルを想定して書かれている． Linux や Mac のユーザーは OS に標準搭載されているターミナルを用いれば良い． Windows を利用している読者は， Windows Subsystem for Linux (WSL) を利用することで，仮想の Linux 環境を構築することを推奨する． Cygwin などの Linux 環境をエミュレートするほかのツールでも構わないが，本書のプログラムは WSL でのみ動作確認を行っている．
    WSL とは， Windows の OS 上で Linux の仮想環境を起動するための， Microsoft 社が公式で提供しているソフトウェアである． Ubuntu など希望の Linux distribution が選択でき，基本的にすべての Linux 向けに作られたプログラム・ソフトウェアを使用することができる．
    執筆時点では WSL 2 が最新版として提供されているので，以下では WSL 2 のインストール手順を簡単に説明する． 細かな詳細などは， 公式ドキュメンテーション を参照のこと．
    前提として，使用される OS は Windows 10 (Pro または Home エディション) でなければならない． さらに，使用している Windows 10 のバージョンが WSL に対応するバージョンであるかを確認する． X64 のシステムでは Version 1903, Build 18362 以上でなければならない． バージョンが対応していない場合は、 Windows のアップデートを行う．
    まず最初に， Administrator 権限で PowerShell を起動する (figure_title)． 左下の Windows メニューの検索バーに powershell と入力すると， PowerShell のプログラムが見つかるはずである， これを右クリックし、 Run as administrator を選択し起動する．
    管理者権限での PowerShell の起動
    PowerShell が起動したら、次のコマンドを実行する．
    powershell
    dism.exe /online /enable-feature /featurename:Microsoft-Windows-Subsystem-Linux /all /norestart
    dism.exe /online /enable-feature /featurename:Microsoft-Windows-Subsystem-Linux /all /norestart
    実行して、“The operation completed successfully.” と出力されるのを確認する． これで WSL が enable される．
    次に，先ほどと同じ Administrator 権限で開いた PowerShell で次のコマンドを実行する。
    powershell
    dism.exe /online /enable-feature /featurename:VirtualMachinePlatform /all /norestart
    dism.exe /online /enable-feature /featurename:VirtualMachinePlatform /all /norestart
    実行して， “The operation completed successfully.” と出力されるのを確認する． これが確認出来たら、一度コンピュータを再起動する．
    続いて， Linux kernel update package を次のリンクからダウンロードする． https://wslstorestorage.blob.core.windows.net/wslblob/wsl_update_x64.msi
    ダウンロードしたファイルをダブルクリックして実行する． ダイアログに従ってインストールを完了させる．
    そうしたら，再び PowerShell を開き次のコマンドを実行する。
    powershell
    wsl --set-default-version 2
    wsl --set-default-version 2
    最後に、自分の好みの Linux distribution をインストールする． ここでは Ubuntu 20.04 をインストールしよう．
    Microsoft store のアプリを起動し，検索バーに Ubuntu と入力する． Ubuntu 20.04 LTS という項目が見つかるはずなので，それを開き， “Get” ボタンをクリックする (figure_title)． しばらく待つと， Ubuntu 20.04 のインストールが完了する．
    Microsoft store から Ubuntu 20.04 をインストール
    Ubuntu 20.04 を初回に起動すると，初期設定が自動で開始され，数分待つことになる． 初期設定が終わると，ユーザー名・パスワードを設定するようプロンプトが出るので，プロンプトに従い入力する．
    これで WSL2 のインストールが完了した． 早速 WSL2 を起動してみよう． 左下の Windows メニューの検索バーに Ubuntu と入力すると， Ubuntu 20.04 のプログラムが見つかるはずである (figure_title)． クリックして起動しよう．
    Ubuntu 20.04 の起動
    すると，ターミナルの黒い画面が立ち上がるだろう (figure_title)． ls, top などのコマンドを打ってみて， WSL がきちんと動作していることを確認しよう．
    WSL の起動画面
    オプションとして， Windows Terminal というマイクロソフトから提供されているツールを使うと，より快適に WSL を使用することができる． 興味のある読者はこちらのインストールも推奨する．
    Docker のインストール ​
    Docker のインストールの方法は OS によって異なる．
    Mac ユーザーは， Docker Desktop をインストールする． インストールの方法は， Docker のウェブサイト から， Mac 版の Docker Desktop をダウンロードし，ダウンロードされたファイルをダブルクリックし， Applications のフォルダにドラッグするだけで良い． 詳細は 公式ドキュメンテーション を参照のこと．
    Windows のユーザーは，Docker Desktop をインストールする． その際， WSL 2 が事前にインストールされていなければならない． 詳細は 公式ドキュメンテーション を参照のこと． Docker Desktop をインストールすると， WSL からも docker コマンドが使用できるようになる．
    Linux ユーザー (特に Ubuntu ユーザー) については，インストールの方法はいくつかのアプローチがある． 公式ドキュメンテーション にいくつかのインストールの方法が示されているので，詳しい情報はそちらを参照いただきたい．
    最も簡単な方法は， Docker が公式で提供しているインストールスクリプトを用いる方法である． この場合，次のコマンドを実行することで Docker がインストールされる．
    shell
    $ curl -fsSL https://get.docker.com -o get-docker.sh
 $ sudo sh get-docker.sh
    $ curl -fsSL https://get.docker.com -o get-docker.sh
-$ sudo sh get-docker.sh
    デフォルトのインストールでは， root ユーザーのみが docker コマンドを使用できる設定になっている． 従って，コマンドには毎回 sudo を付け加える必要がある． これが面倒だと感じる場合は，次のステップにより，使用するユーザーを docker というグループに追加する (詳細は 公式ドキュメンテーション "Post-installation steps for Linux" を参照)．
    まず最初に， docker という名前にグループを追加する． インストールによっては，既に docker グループが作られている場合もある．
    shell
    $ sudo groupadd docker
    $ sudo groupadd docker
    次に，現在使用しているユーザーを docker グループに加える．
    shell
    $ sudo usermod -aG docker $USER
    $ sudo usermod -aG docker $USER
    ここまでできたら，一度ログアウトし，再度ログインする． これによって，グループの変更がターミナルのセッションに反映される．
    設定が正しくできているかを確認するため，次のコマンドを実行してみる．
    shell
    $ docker run hello-world
    $ docker run hello-world
    sudo なしでコンテナが実行できたならば，設定は完了である．
    Python venv クイックガイド ​
    他人からもらったプログラムで， numpy や scipy のバージョンが違う！などの理由で，プログラムが動かない，という経験をしたことがある人は多いのではないだろうか． もし，自分の計算機の中に一つしか Python 環境がないとすると，プロジェクトを切り替えるごとに正しいバージョンをインストールし直さなければならず，これは大変な手間である．
    コードのシェアをよりスムーズにするためには，ライブラリのバージョンはプロジェクトごとに管理されるべきである． それを可能にするのが Python 仮想環境とよばれるツールであり， venv, pyenv, conda などがよく使われる．
    そのなかでも， venv は Python に標準搭載されているのでとても便利である． pyenv や conda は，別途インストールの必要があるが，それぞれの長所もある．
    venv を使って仮想環境を作成するには，
    shell
    $ python -m venv .env
    $ python -m venv .env
    と実行する． これにより .env/ というディレクトリが作られ，このディレクトリに依存するライブラリが保存されることになる．
    この新たな仮想環境を起動するには
    shell
    $ source .env/bin/activate
    $ source .env/bin/activate
    と実行する．
    シェルのプロンプトに (.env) という文字が追加されていることを確認しよう (figure_title)． これが， "いまあなたは venv の中にいますよ" というしるしになる．
    venv を起動したときのプロンプト
    仮想環境を起動すると，それ以降実行する pip コマンドは， .env/ 以下にインストールされる．このようにして，プロジェクトごとに使うライブラリのバージョンを切り分けることができる．
    Python では requirements.txt というファイルに依存ライブラリを記述するのが一般的な慣例である．他人からもらったプログラムに， requirements.txt が定義されていれば，
    shell
    $ pip install -r requirements.txt
    $ pip install -r requirements.txt
    と実行することで，必要なライブラリをインストールし，瞬時に Python 環境を再現することができる．
    venv による仮想環境を保存するディレクトリの名前は任意に選べることができるが， .env という名前を用いるのが一般的である．
    ハンズオン実行用の Docker image の使い方 ​
    ハンズオンを実行するために必要な， Node.js, Python, AWS CDK などがインストールされた Docker image を用意した． これを使用することで，自分のローカルマシンに諸々をインストールする必要なく，すぐにハンズオンのコードが実行できる．
    ハンズオンのいくつかのコマンドは Docker の外 = ローカルマシンのリアル環境で実行されなければならない． それらについてはハンズオンの該当箇所に注意書きとして記してある．
    Docker イメージは Docker Hub においてある． Docker イメージのビルドファイルは GitHub の docker/Dockerfile にある．
    次のコマンドでコンテナを起動する．
    shell
    $ docker run -it tomomano/labc:latest
    $ docker run -it tomomano/labc:latest
    初回にコマンドを実行したときのみ，イメージが Docker Hub からダウンロード (pull) される． 二回目以降はローカルにダウンロードされたイメージが使用される．
    コンテナが起動すると，次のようなインタラクティブシェルが表示されるはずである (起動時に -it のオプションをつけたのがポイントである)．
    root@aws-handson:~$</programlisting>
    この状態で ls コマンドを打つと， handson/ というディレクトリがあるはずである． ここに cd する．
    shell
    $ cd handson
    $ cd handson
    すると，各ハンズオンごとのディレクトリが見つかるはずである．
    あとは，ハンズオンごとにディレクトリを移動し，ハンズオンごとの virtualenv を作成し，スタックのデプロイを行えばよい (??? など参照)． ハンズオンごとに使用する依存ライブラリが異なるので，それぞれのハンズオンごとに virtualenv を作成するという設計になっている．
    AWS の認証情報を設定することも忘れずに． AWS CLI のインストール で記述したように， AWS_ACCESS_KEY_ID などの環境変数を設定するのが簡単な方法である． あるいは，ローカルマシンの ~/.aws/credentials に認証情報が書き込まれているなら，このディレクトリをコンテナにマウントすることで，同じ認証ファイルをコンテナ内部から参照することが可能である． この選択肢を取る場合は，次のコマンドでコンテナを起動する．
    shell
    $ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
    $ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
    これにより，ローカルマシンの ~/.aws をコンテナの /root/.aws にマウントすることができる． 最後の :ro は read-only を意味する． 大切な認証ファイルが誤って書き換えられてしまわないように， read-only のフラグをつけることをおすすめする．
    /root/ がコンテナ環境におけるホームディレクトリである． ここで紹介した認証ファイルをマウントするテクニックは， SSH 鍵をコンテナに渡すときなどにも使える．

-    
+$ sudo sh get-docker.sh
デフォルトのインストールでは， root ユーザーのみが docker コマンドを使用できる設定になっている． 従って，コマンドには毎回 sudo を付け加える必要がある． これが面倒だと感じる場合は，次のステップにより，使用するユーザーを docker というグループに追加する (詳細は 公式ドキュメンテーション "Post-installation steps for Linux" を参照)．
まず最初に， docker という名前にグループを追加する． インストールによっては，既に docker グループが作られている場合もある．
shell
$ sudo groupadd docker
$ sudo groupadd docker
次に，現在使用しているユーザーを docker グループに加える．
shell
$ sudo usermod -aG docker $USER
$ sudo usermod -aG docker $USER
ここまでできたら，一度ログアウトし，再度ログインする． これによって，グループの変更がターミナルのセッションに反映される．
設定が正しくできているかを確認するため，次のコマンドを実行してみる．
shell
$ docker run hello-world
$ docker run hello-world
sudo なしでコンテナが実行できたならば，設定は完了である．
Python venv クイックガイド ​
他人からもらったプログラムで， numpy や scipy のバージョンが違う！などの理由で，プログラムが動かない，という経験をしたことがある人は多いのではないだろうか． もし，自分の計算機の中に一つしか Python 環境がないとすると，プロジェクトを切り替えるごとに正しいバージョンをインストールし直さなければならず，これは大変な手間である．
コードのシェアをよりスムーズにするためには，ライブラリのバージョンはプロジェクトごとに管理されるべきである． それを可能にするのが Python 仮想環境とよばれるツールであり， venv, pyenv, conda などがよく使われる．
そのなかでも， venv は Python に標準搭載されているのでとても便利である． pyenv や conda は，別途インストールの必要があるが，それぞれの長所もある．
venv を使って仮想環境を作成するには，
shell
$ python -m venv .env
$ python -m venv .env
と実行する． これにより .env/ というディレクトリが作られ，このディレクトリに依存するライブラリが保存されることになる．
この新たな仮想環境を起動するには
shell
$ source .env/bin/activate
$ source .env/bin/activate
と実行する．
シェルのプロンプトに (.env) という文字が追加されていることを確認しよう (figure_title)． これが， "いまあなたは venv の中にいますよ" というしるしになる．
venv を起動したときのプロンプト
仮想環境を起動すると，それ以降実行する pip コマンドは， .env/ 以下にインストールされる．このようにして，プロジェクトごとに使うライブラリのバージョンを切り分けることができる．
Python では requirements.txt というファイルに依存ライブラリを記述するのが一般的な慣例である．他人からもらったプログラムに， requirements.txt が定義されていれば，
shell
$ pip install -r requirements.txt
$ pip install -r requirements.txt
と実行することで，必要なライブラリをインストールし，瞬時に Python 環境を再現することができる．
venv による仮想環境を保存するディレクトリの名前は任意に選べることができるが， .env という名前を用いるのが一般的である．
ハンズオン実行用の Docker image の使い方 ​
ハンズオンを実行するために必要な， Node.js, Python, AWS CDK などがインストールされた Docker image を用意した． これを使用することで，自分のローカルマシンに諸々をインストールする必要なく，すぐにハンズオンのコードが実行できる．
ハンズオンのいくつかのコマンドは Docker の外 = ローカルマシンのリアル環境で実行されなければならない． それらについてはハンズオンの該当箇所に注意書きとして記してある．
Docker イメージは Docker Hub においてある． Docker イメージのビルドファイルは GitHub の docker/Dockerfile にある．
次のコマンドでコンテナを起動する．
shell
$ docker run -it tomomano/labc:latest
$ docker run -it tomomano/labc:latest
初回にコマンドを実行したときのみ，イメージが Docker Hub からダウンロード (pull) される． 二回目以降はローカルにダウンロードされたイメージが使用される．
コンテナが起動すると，次のようなインタラクティブシェルが表示されるはずである (起動時に -it のオプションをつけたのがポイントである)．
root@aws-handson:~$</programlisting>
この状態で ls コマンドを打つと， handson/ というディレクトリがあるはずである． ここに cd する．
shell
$ cd handson
$ cd handson
すると，各ハンズオンごとのディレクトリが見つかるはずである．
あとは，ハンズオンごとにディレクトリを移動し，ハンズオンごとの virtualenv を作成し，スタックのデプロイを行えばよい (??? など参照)． ハンズオンごとに使用する依存ライブラリが異なるので，それぞれのハンズオンごとに virtualenv を作成するという設計になっている．
AWS の認証情報を設定することも忘れずに． AWS CLI のインストール で記述したように， AWS_ACCESS_KEY_ID などの環境変数を設定するのが簡単な方法である． あるいは，ローカルマシンの ~/.aws/credentials に認証情報が書き込まれているなら，このディレクトリをコンテナにマウントすることで，同じ認証ファイルをコンテナ内部から参照することが可能である． この選択肢を取る場合は，次のコマンドでコンテナを起動する．
shell
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
これにより，ローカルマシンの ~/.aws をコンテナの /root/.aws にマウントすることができる． 最後の :ro は read-only を意味する． 大切な認証ファイルが誤って書き換えられてしまわないように， read-only のフラグをつけることをおすすめする．
/root/ がコンテナ環境におけるホームディレクトリである． ここで紹介した認証ファイルをマウントするテクニックは， SSH 鍵をコンテナに渡すときなどにも使える．

+    
     
   
 
\ No newline at end of file
diff --git a/development/aws/assignments.html b/development/aws/assignments.html
index 94c7393a..b001926a 100644
--- a/development/aws/assignments.html
+++ b/development/aws/assignments.html
@@ -5,12 +5,12 @@
     
     課題 | Toshiki's Note
     
-    
+    
     
     
   
   
-  
+  
   
   
   
@@ -32,9 +32,9 @@
   
   
   
-    
Toshiki's Note
Appearance
TOC
課題 ​
以下の課題を提出．

-    
+    
Toshiki's Note
Appearance
TOC
課題 ​
以下の課題を提出．

+    
     
   
 
\ No newline at end of file
diff --git a/development/aws/author.html b/development/aws/author.html
index 15fc463b..93e7f22b 100644
--- a/development/aws/author.html
+++ b/development/aws/author.html
@@ -5,12 +5,12 @@
     
     著者紹介 | Toshiki's Note
     
-    
+    
     
     
   
   
-  
+  
   
   
   
@@ -32,9 +32,9 @@
   
   
   
-    
Toshiki's Note
Appearance
TOC
著者紹介 ​
真野 智之 (Tomoyuki Mano)
情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．
連絡先　tomoyukimano@gmail.com
GitHub https://github.com/tomomano

-    
+    
Toshiki's Note
Appearance
TOC
著者紹介 ​
真野 智之 (Tomoyuki Mano)
情報理工学博士 (東京大学大学院情報理工学系研究科システム情報学専攻)． 2021 年より日本学術振興会特別研究員 (PD) (現職)． 沖縄科学技術大学院大学 (OIST) にてポスドク研究員として働く． 現在の研究分野は神経科学・神経情報学． 趣味は料理・ランニング・鉄道・アニメ，村上春樹の熱烈な愛読家．
連絡先　tomoyukimano@gmail.com
GitHub https://github.com/tomomano

+    
     
   
 
\ No newline at end of file
diff --git a/development/aws/aws-batch.html b/development/aws/aws-batch.html
index b5fe733c..8eb4e6ac 100644
--- a/development/aws/aws-batch.html
+++ b/development/aws/aws-batch.html
@@ -5,13 +5,13 @@
     
     Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する | Toshiki's Note
     
-    
+    
     
     
   
   
   
-  
+  
   
   
   
@@ -488,9 +488,9 @@
 for i in range(3):
     for j in range(3):
         text = ax.text(j+0.5, i+0.5, f"{grid[i, j]:0.1f}",
-                       ha="center", va="center", color="w")
最終的に出力されるプロットが figure_title である．
ハイパーパラメータのグリッドサーチの結果
このプロットから，差は僅かであるが，学習率が 0.1 のときに精度は最大となることがわかる． また，学習率 0.1 のときはモメンタムを変えても大きな差は生じないことが見て取れる．
今回のパラメータサーチは勉強用に極めて単純化されたものである点は承知いただきたい．
たとえば，今回は学習率が 0.1 が最も良いとされたが，それは訓練のエポックを 100 に限定しているからかもしれない． 学習率が低いとその分訓練に必要なエポック数も多くなる． 訓練のエポック数をもっと増やせばまた違った結果が観察される可能性はある．
また，今回は MNIST の訓練データ 60,000 枚のうち， 48,000 枚を訓練データ，残り 12,000 枚を検証データとして用いた． この分割は乱数を固定してランダムに行ったが，もしこの分割によるデータのバイアスを気にするならば， k 個の異なる学習・検証データの分割をあらかじめ用意して，複数回モデルの評価を行う (k-fold cross-validation) 方法も，より精緻なアプローチとして考えられる．
以上のようにして， CNN を用いた MNIST 分類モデルのハイパーパラメータの最適化の一連の流れを体験した． AWS Batch を利用することで，比較的少ないプログラミングで，動的に EC2 クラスターを制御し，並列にジョブを処理するシステムが構築できた． ここまで EC2 を使いこなすことができれば，多くの問題を自力で解くことが可能になるだろう！
スタックの削除 ​
これにて，本ハンズオンは終了である．最後にスタックを削除しよう． 今回のスタックを削除するにあたり，ECR に配置された Docker のイメージは手動で削除されなければならない (これをしないと， cdk destroy を実行したときにエラーになってしまう． これは CloudFormation の仕様なので従うしかない)．
ECR の Docker image を削除するには， ECR のコンソールに行き，イメージが配置されたレポジトリを開く． そして，画面右上の DELETE ボタンを押して削除する (figure_title)．
ECR から Docker image を削除する
あるいは， AWS CLI から同様の操作を行うには，以下のコマンドを用いる (XXXX は自分の ECR レポジトリ名に置き換える)．
shell
$ aws ecr batch-delete-image --repository-name XXXX --image-ids imageTag=latest
$ aws ecr batch-delete-image --repository-name XXXX --image-ids imageTag=latest
image の削除が完了したうえで，次のコマンドでスタックを削除する．
shell
$ cdk destroy
$ cdk destroy
??? === クラウドを用いた機械学習アプリケーションの開発とデバッグ
本章で紹介したハンズオンでは， AWS Batch を使用することでニューラルネットの学習を複数並列に実行し，高速化を実現した． 本章の最後の話題として，クラウドを用いた機械学習アプリケーションの開発とデバッグの方法について述べよう．
ローカルに GPU を搭載した強力なマシンがなく，クラウドを利用する予算が確保されているのであれば， figure_title のような開発のスキームが理想的であると考える． 最初の段階では， ??? で見たような方法で， GPU 搭載型の EC2 インスタンスを作成し， Jupyter Notebook などのインタラクティブな環境で様々なモデルを試し実験を行う． Jupyter である程度アプリケーションが完成してきたタイミングで，作成したアプリケーションを Docker イメージにパッケージングする． そして， EC2 上で docker run を行い，作成したイメージがバグなく動作するか確認を行う． その次に，ハイパーパラメータの最適化などのチューニングを， Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する のハンズオンで学んだ AWS Batch などの計算システムを利用して行う． よい深層学習モデルが完成したら，仕上げに大規模データへの推論処理を行うシステムを ??? を参考に構築する．
実際，本書ではこの流れに沿って演習を進めてきた． MNIST タスクを解くモデルを，最初 Jupyter Notebook を使用して実験し，そのコードをほとんどそのまま Docker にパッケージし， AWS Batch を用いてハイパーパラメータサーチを行った． このサイクルを繰り返すことで，クラウドを最大限に活用した機械学習アプリケーションの開発を進めることができる．
クラウドを活用した機械学習アプリケーションの開発フロー
小括 ​
ここまでが，本書第二部の内容である． 第一部に引き続き盛りだくさんの内容であったが，ついてこれたであろうか？
第二部ではまず最初に，深層学習の計算をクラウドで実行するため， GPU 搭載型の EC2 インスタンスの起動について解説した． さらに，ハンズオンでは，クラウドに起動した仮想サーバーを使って MNIST 文字認識タスクを解くニューラルネットを訓練した (???)．
また，より大規模な機械学習アプリケーションを作るための手段として， Docker と ECS によるクラスターの初歩を説明した (???)． その応用として，英語で与えられた文章問題への回答を自動で生成するボットをクラウドに展開した (???)． タスクの投入に応じて動的に計算リソースが作成・削除される様子を実際に体験できただろう．
さらに， Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する では AWS Batch を用いてニューラルネットの学習を並列に実行する方法を紹介した． ここで紹介した方法は，ミニマムであるが，計算機システムを大規模化していくためのエッセンスが網羅されている． これらのハンズオン体験から，クラウド技術を応用してどのように現実世界の問題を解いていくのか，なんとなくイメージが伝わっただろうか？
本書の第三部では，さらにレベルアップし，サーバーレスアーキテクチャという最新のクラウドの設計手法について解説する． その応用として，ハンズオンでは簡単な SNS サービスをゼロから実装する． 引き続きクラウドの最先端の世界を楽しんでいこう！
- + ha="center", va="center", color="w")

最終的に出力されるプロットが figure_title である．

ハイパーパラメータのグリッドサーチの結果

このプロットから，差は僅かであるが，学習率が 0.1 のときに精度は最大となることがわかる． また，学習率 0.1 のときはモメンタムを変えても大きな差は生じないことが見て取れる．

今回のパラメータサーチは勉強用に極めて単純化されたものである点は承知いただきたい．

たとえば，今回は学習率が 0.1 が最も良いとされたが，それは訓練のエポックを 100 に限定しているからかもしれない． 学習率が低いとその分訓練に必要なエポック数も多くなる． 訓練のエポック数をもっと増やせばまた違った結果が観察される可能性はある．

また，今回は MNIST の訓練データ 60,000 枚のうち， 48,000 枚を訓練データ，残り 12,000 枚を検証データとして用いた． この分割は乱数を固定してランダムに行ったが，もしこの分割によるデータのバイアスを気にするならば， k 個の異なる学習・検証データの分割をあらかじめ用意して，複数回モデルの評価を行う (k-fold cross-validation) 方法も，より精緻なアプローチとして考えられる．

以上のようにして， CNN を用いた MNIST 分類モデルのハイパーパラメータの最適化の一連の流れを体験した． AWS Batch を利用することで，比較的少ないプログラミングで，動的に EC2 クラスターを制御し，並列にジョブを処理するシステムが構築できた． ここまで EC2 を使いこなすことができれば，多くの問題を自力で解くことが可能になるだろう！

スタックの削除 ​

これにて，本ハンズオンは終了である．最後にスタックを削除しよう． 今回のスタックを削除するにあたり，ECR に配置された Docker のイメージは手動で削除されなければならない (これをしないと， cdk destroy を実行したときにエラーになってしまう． これは CloudFormation の仕様なので従うしかない)．

ECR の Docker image を削除するには， ECR のコンソールに行き，イメージが配置されたレポジトリを開く． そして，画面右上の DELETE ボタンを押して削除する (figure_title)．

ECR から Docker image を削除する

あるいは， AWS CLI から同様の操作を行うには，以下のコマンドを用いる (XXXX は自分の ECR レポジトリ名に置き換える)．

shell
$ aws ecr batch-delete-image --repository-name XXXX --image-ids imageTag=latest
$ aws ecr batch-delete-image --repository-name XXXX --image-ids imageTag=latest

image の削除が完了したうえで，次のコマンドでスタックを削除する．

shell
$ cdk destroy
$ cdk destroy

??? === クラウドを用いた機械学習アプリケーションの開発とデバッグ

本章で紹介したハンズオンでは， AWS Batch を使用することでニューラルネットの学習を複数並列に実行し，高速化を実現した． 本章の最後の話題として，クラウドを用いた機械学習アプリケーションの開発とデバッグの方法について述べよう．

ローカルに GPU を搭載した強力なマシンがなく，クラウドを利用する予算が確保されているのであれば， figure_title のような開発のスキームが理想的であると考える． 最初の段階では， ??? で見たような方法で， GPU 搭載型の EC2 インスタンスを作成し， Jupyter Notebook などのインタラクティブな環境で様々なモデルを試し実験を行う． Jupyter である程度アプリケーションが完成してきたタイミングで，作成したアプリケーションを Docker イメージにパッケージングする． そして， EC2 上で docker run を行い，作成したイメージがバグなく動作するか確認を行う． その次に，ハイパーパラメータの最適化などのチューニングを， Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する のハンズオンで学んだ AWS Batch などの計算システムを利用して行う． よい深層学習モデルが完成したら，仕上げに大規模データへの推論処理を行うシステムを ??? を参考に構築する．

実際，本書ではこの流れに沿って演習を進めてきた． MNIST タスクを解くモデルを，最初 Jupyter Notebook を使用して実験し，そのコードをほとんどそのまま Docker にパッケージし， AWS Batch を用いてハイパーパラメータサーチを行った． このサイクルを繰り返すことで，クラウドを最大限に活用した機械学習アプリケーションの開発を進めることができる．

クラウドを活用した機械学習アプリケーションの開発フロー

小括 ​

ここまでが，本書第二部の内容である． 第一部に引き続き盛りだくさんの内容であったが，ついてこれたであろうか？

第二部ではまず最初に，深層学習の計算をクラウドで実行するため， GPU 搭載型の EC2 インスタンスの起動について解説した． さらに，ハンズオンでは，クラウドに起動した仮想サーバーを使って MNIST 文字認識タスクを解くニューラルネットを訓練した (???)．

また，より大規模な機械学習アプリケーションを作るための手段として， Docker と ECS によるクラスターの初歩を説明した (???)． その応用として，英語で与えられた文章問題への回答を自動で生成するボットをクラウドに展開した (???)． タスクの投入に応じて動的に計算リソースが作成・削除される様子を実際に体験できただろう．

さらに， Hands-on #4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する では AWS Batch を用いてニューラルネットの学習を並列に実行する方法を紹介した． ここで紹介した方法は，ミニマムであるが，計算機システムを大規模化していくためのエッセンスが網羅されている． これらのハンズオン体験から，クラウド技術を応用してどのように現実世界の問題を解いていくのか，なんとなくイメージが伝わっただろうか？

本書の第三部では，さらにレベルアップし，サーバーレスアーキテクチャという最新のクラウドの設計手法について解説する． その応用として，ハンズオンでは簡単な SNS サービスをゼロから実装する． 引き続きクラウドの最先端の世界を楽しんでいこう！

+ \ No newline at end of file diff --git a/development/aws/aws-get-started.html b/development/aws/aws-get-started.html index 4b4071e1..d272f9d3 100644 --- a/development/aws/aws-get-started.html +++ b/development/aws/aws-get-started.html @@ -5,14 +5,14 @@ AWS 入門 | Toshiki's Note - + - + @@ -182,9 +182,9 @@ machine_image=ec2.MachineImage.latest_amazon_linux(), vpc=vpc, ... - )

このコードは，一つ前に示した JSON を使った CloudFormation と実質的に同じことを記述している． とても煩雑だった CloudFormation ファイルに比べて， CDK と Python を使うことで格段に短く，わかりやすく記述できることができるのがわかるだろう．

本書の主題は，CDK を使って，コードを書きながら AWS の概念や開発方法を学んでいくことである． 後の章では CDK を使って様々なハンズオンを実施していく． 早速，最初のハンズオンでは， CDK を使って EC2 インスタンスを作成する方法を学んでいこう．

  • AWS CDK Examples: CDK を使ったプロジェクトの例が多数紹介されている． ここにある例をテンプレートに自分のアプリケーションの開発を進めるとよい．
- + )

このコードは，一つ前に示した JSON を使った CloudFormation と実質的に同じことを記述している． とても煩雑だった CloudFormation ファイルに比べて， CDK と Python を使うことで格段に短く，わかりやすく記述できることができるのがわかるだろう．

本書の主題は，CDK を使って，コードを書きながら AWS の概念や開発方法を学んでいくことである． 後の章では CDK を使って様々なハンズオンを実施していく． 早速，最初のハンズオンでは， CDK を使って EC2 インスタンスを作成する方法を学んでいこう．

  • AWS CDK Examples: CDK を使ったプロジェクトの例が多数紹介されている． ここにある例をテンプレートに自分のアプリケーションの開発を進めるとよい．
+ \ No newline at end of file diff --git a/development/aws/closing.html b/development/aws/closing.html index 061f9538..4a533915 100644 --- a/development/aws/closing.html +++ b/development/aws/closing.html @@ -5,12 +5,12 @@ まとめ | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

まとめ ​

- +
Toshiki's Note

Appearance

TOC

まとめ ​

+ \ No newline at end of file diff --git a/development/aws/cloud.html b/development/aws/cloud.html index 6261a3bf..d4c88405 100644 --- a/development/aws/cloud.html +++ b/development/aws/cloud.html @@ -5,13 +5,13 @@ クラウド概論 | Toshiki's Note - + - + @@ -33,9 +33,9 @@ -
Toshiki's Note

Appearance

TOC

クラウド概論 ​

クラウドとは？ ​

Cloud

クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．

学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．

The NIST Definition of Cloud Computing

これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．

  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．

  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．

  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．

  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．

  • Measured service 計算資源の利用量を計測・監視することができる．

…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．

個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．

クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．

このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．

著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．

クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．

先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．

  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．

  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．

  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．

本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．

SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (???) などの演習を提供する．

なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については ??? などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．

最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．

一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．

第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．

本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．

なぜクラウドを使うのか？ ​

上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．

  1. 自由にサーバーのサイズをスケールできる

    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない

    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0

    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．

    クラウドと自前計算機環境の経済的コストの比較

とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．

ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．

  1. クラウドは賢く使わないといけない

    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ

    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ

    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．

Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？

Terminal

この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．

時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．

最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．

- +
Toshiki's Note

Appearance

TOC

クラウド概論 ​

クラウドとは？ ​

Cloud

クラウドとはなにか？ クラウドという言葉は，それ自身がとても広い意味をもつので，厳密な定義付けを行うことは難しい．

学術的な意味でのクラウドの定義づけをするとしたら，NIST(米国・国立標準技術研究所) による The NIST Definition of Cloud Computing が引用されることが多い． ここに記載されたクラウドの定義・モデルを図示したのが figure_title である．

The NIST Definition of Cloud Computing

これによると，クラウドとは以下の要件が満たされたハードウェア/ソフトウェアの総体のことをいう．

  • On-demand self-service 利用者のリクエストに応じて計算資源が自動的に割り当てられる．

  • Broad network access 利用者はネットワークを通じてクラウドにアクセスできる．

  • Resource pooling クラウドプロバイダーは，所有する計算資源を分割することで複数の利用者に計算資源を割り当てる．

  • Rapid elasticity 利用者のリクエストに応じて，迅速に計算資源の拡大あるいは縮小を行うことができる．

  • Measured service 計算資源の利用量を計測・監視することができる．

…と，いわれても抽象的でよくわからないかもしれない．もう少し具体的な話をする．

個人が所有する計算機で， CPU をアップグレードしようと思ったら，物理的に筐体を開け，CPU ソケットを露出させ，新しい CPU に交換する必要があるだろう． あるいは，ストレージがいっぱいになってしまったら，古いディスクを抜き取り，新しいディスクを挿入する必要がある． 計算機の場所を移動させたときには，新しい部屋の LAN ケーブルを差し込まないとネットワークには接続できない．

クラウドでは，これらの操作がプログラムからのコマンドによって実行できる． CPU が 1000 個欲しいと思ったならば，そのようにクラウドプロバイダーにリクエストを送れば良い． すると，数分もしないうちに 1000 CPU の計算資源が割り当てられる． ストレージを 1TB から 10TB に拡張しようと思ったならば，そのようにコマンドを送ればよい (これは，Google Drive や Dropbox などのサービスなどで馴染みのある人も多いだろう)． 計算資源を使い終わったら，そのことをプロバイダーに伝えれば，割り当て分はすぐさま削除される． クラウドプロバイダーは，使った計算資源の量を正確にモニタリングしており，その量をもとに利用料金の計算が行われる．

このように，クラウドの本質は物理的なハードウェアの仮想化・抽象化であり，利用者はコマンドを通じて，まるでソフトウェアの一部かのように，物理的なハードウェアの管理・運用を行うことができる． もちろん，背後では，データセンターに置かれた膨大な数の計算機が大量の電力を消費しながら稼働している． クラウドプロバイダーはデータセンターの計算資源を上手にやりくりし，ソフトウェアとしてのインターフェースをユーザーに提供することで，このような仮想化・抽象化を達成しているわけである． クラウドプロバイダーの視点からすると，大勢のユーザーに計算機を貸し出し，データセンターの稼働率を常時 100%に近づけることで，利益率の最大化を図っているのである．

著者の言葉で，クラウドの重要な特性を定義するならば，以下のようになる．

クラウドとは計算機ハードウェアの抽象化である．つまり，物理的なハードウェアをソフトウェアの一部かのように自在に操作・拡大・接続することを可能にする技術である．

先述の The NIST Definition of Cloud Computing に戻ると，クラウドプロバイダーによるクラウドサービスの形態としては，次の三つが定義されている (figure_title)．

  • Software as a Service (SaaS): クラウド上で実行されるアプリケーションをサービスとして利用者に提供する形態． 例として， Google Drive や Slack などが挙げられる． 利用者は，背後にあるクラウドのインフラ (ネットワークやサーバーなど) には直接触れず，アプリケーションとして提供されているクラウドサービスを享受する．

  • Platform as a Service (PaaS): 顧客の作成したアプリケーション (多くの場合データベースと API リクエスト処理を行うサーバーのコードから構成される) をデプロイする環境をサービスとして利用者に提供する形態． PaaS では利用者はクラウドのインフラに直接触れることはなく，計算負荷が増減した際のサーバーのスケーリングはクラウドプロバイダーによってなされる． 例としては， Google App Engine や Heroku などがある．

  • Infrastructure as a Service (IaaS): クラウド上の計算インフラストラクチャーを従量課金制で利用者に提供する形態． 利用者は必要なネットワーク・サーバー・ストレージをプロバイダーから借り受け，そこに自身のアプリケーションを展開し運用する． IaaS の例としては AWS EC2 などが挙げられる．

本書が扱うのは，主に IaaS におけるクラウド開発である． すなわち，開発者がクラウドのインフラを直接操作し，所望のネットワーク・サーバー・ストレージなどを一から構成し，そこにアプリケーションを展開するというクラウド開発である． この意味において，クラウドの開発とはクラウドインフラストラクチャーを定義・展開するプログラムを構築するステップとインフラ上で実際に走るアプリケーションを作成するステップの二つに分けることができる． この二つは，プログラマーの技術としてはある程度分業を行うことが可能であるが，最も効率化・最適化されたクラウドシステムを構築するためには両方の理解が必須である． 本書では，前者 (クラウドインフラの記述) に重きを置きつつ，アプリケーションレイヤーの話題も取り扱う． PaaS とは，開発者はアプリケーションレイヤーの開発に注力し，クラウドインフラの部分はクラウドプロバイダーに依存するという概念である． PaaS は，クラウドインフラの開発が不要になることで開発の時間が短縮されるが，細かなインフラの挙動はコントロールできないという限界がある． 本書では PaaS についてはとくに取り扱わない．

SaaS は本書の文脈では開発による"成果物"と捉えられるだろう． すなわち， IaaS を構成するプログラムを作成し展開することによって，一般の人が利用できるようなウェブ上の計算サービスやデータベースを提供することが開発の最終ゴールである． 本書のハンズオンではその実例として，シンプルな SNS の作成 (???) などの演習を提供する．

なお，最近では Function as a Service (FaaS) やサーバーレスコンピューティングなども新たなクラウドのカテゴリとして認知されている． これらの概念については ??? などの章で詳しく触れていく． 本書を読み進める中で明らかになるように，クラウドの技術は日進月歩である． 本書では実用的・教育的な観点から，従来的なクラウドの設計概念に触れたあと，サーバーレスなどの最新の技術も網羅するので，楽しみにしながら読み進めていただきたい．

最後に，The NIST Definition of Cloud Computing によると，クラウドの運用形態について次のような定義がなされている (figure_title)． 特定の組織・団体・企業の内部のみで使用されるクラウドを，プライベートクラウド (private cloud) とよぶ． 例えば，大学や研究機関では，その機関の構成員向けの大規模計算機サーバーが運用されていることが多い． プライベートクラウドは，組織の構成員ならば無料もしくは極めて割安のコストで計算を実行できる． しかし，使用できる計算資源の上限は限られる場合が多く，拡張時の柔軟性に欠ける場合もある．

一方，商用のサービスとして一般の顧客に向けたクラウドのことを，パブリッククラウド (public cloud) とよぶ． 有名なパブリッククラウドプラットフォームの例を挙げると， Google 社が提供する Google Cloud Platform (GCP)， Microsoft 社が提供する Azure， Amazon 社が提供する Amazon Web Services (AWS) などがある． パブリッククラウドを利用する場合は，プロバイダーの設定した利用料金を支払うことになる． その分，巨大なデータセンターを運用する企業の計算資源にアクセスすることができるので，計算のキャパシティは無尽蔵にあるといって過言でない．

第三のクラウドの運用形態として，コミュニティクラウド (community cloud) が挙げられる． これは，例えば政府の省庁・機関など目的・役割を共有する団体・組織が共有して運用するクラウドを指す． 最後に，ハイブリッドクラウド (hybrid cloud) という形態もあり，これはプライベート・パブリック・コミュニティクラウドの二つ以上の組み合わせによって構成されるクラウドのことである． データ保護の観点から，いくつかの機密データやプライバシーに関わる情報はプライベートクラウドに保持し，残りのシステムをパブリッククラウドに依存する，などの形態が想定される．

本書で説明するのは，基本的にパブリッククラウドを使ったクラウド開発である． 特に，Amazon Web Services (AWS) を使用して，具体的な技術と概念を学んでいく． 一方で，サーバーのスケーリングや仮想計算環境などのテクニックはすべてのクラウドに共通な概念であるので，クラウドのプラットフォームが変わろうと一般に通用する知識も同時に身につくはずだ．

なぜクラウドを使うのか？ ​

上述のように，クラウドとはプログラムを通じて自由に計算資源を操作することのできる計算環境である． ここでは，リアルなローカル計算環境と比べて，なぜクラウドを使うと良いことがあるのかについて述べたい．

  1. 自由にサーバーのサイズをスケールできる

    なにか新しいプロジェクトを始めるとき，あらかじめ必要なサーバーのスペックを知るのは難しい． いきなり大きなサーバーを買うのはリスクが高い． 一方で，小さすぎるサーバーでは，後のアップグレードが面倒である． クラウドを利用すれば，プロジェクトを進めながら，必要な分だけの計算資源を確保することができる． 2. 自分でサーバーをメンテナンスする必要がない

    悲しいことに，コンピュータとは古くなるものである．最近の技術の進歩の速度からすると，5 年も経てば，もはや当時の最新コンピュータも化石と同じである． 5 年ごとにサーバーを入れ替えるのは相当な手間である． またサーバーの停電や故障など不意の障害への対応も必要である． クラウドでは，そのようなインフラの整備やメンテナンスはプロバイダーが自動でやってくれるので，ユーザーが心配する必要がない． 3. 初期コスト 0

    自前の計算環境とクラウドの，経済的なコストのイメージを示したのが figure_title である． クラウドを利用する場合の初期コストは基本的に 0 である． その後，使った利用量に応じてコストが増大していく． 一方，自前の計算環境では，大きな初期コストが生じる． その分，初期投資後のコストの増加は，電気利用料やサーバー維持費などに留まるため，クラウドを利用した場合よりも傾きは小さくなる． 自前の計算機では，ある一定期間後，サーバーのアップグレードなどによる支出が生じることがある． 一方，クラウドを利用する場合は，そのような非連続なコストの増大は基本的に生じない． クラウドのコストのカーブが，自前計算環境のコストのカーブの下にある範囲においては，クラウドを使うことは経済的なコスト削減につながる．

    クラウドと自前計算機環境の経済的コストの比較

とくに，**1.**の点は研究の場面では重要であると筆者は感じる． 研究をやっていて，四六時中計算を走らせ続けるという場合は少ない． むしろ，新しいアルゴリズムが完成したとき・新しいデータが届いたとき，集中的・突発的に計算タスクが増大することが多いだろう． そういったときに，フレキシブルに計算力を増強させることができるのは，クラウドを使う大きなメリットである．

ここまでクラウドを使うメリットを述べたが，逆に，デメリットというのも当然存在する．

  1. クラウドは賢く使わないといけない

    figure_title で示したコストのカーブにあるとおり，使い方によっては自前の計算環境のほうがコスト的に有利な場面は存在しうる． クラウドを利用する際は，使い終わった計算資源はすぐに削除するなど，利用者が賢く管理を行う必要があり，これを怠ると思いもしない額の請求が届く可能性がある． 2. セキュリティ

    クラウドは，インターネットを通じて世界のどこからでもアクセスできる状態にあり，セキュリティ管理を怠ると簡単にハッキングの対象となりうる． ハッキングを受けると，情報流出だけでなく，経済的な損失を被る可能性がある． 3. ラーニングカーブ

    上記のように，コスト・セキュリティなど，クラウドを利用する際に留意しなければならない点は多い． 賢くクラウドを使うには，十分なクラウドの理解が必要であり，そのラーニングカーブを乗り越える必要がある．

Mac/Linux などでコマンドを入力するときに使用する，あの黒い画面のことを Terminal とよんだりする． この言葉の語源をご存知だろうか？

Terminal

この言葉の語源は，コンピュータが誕生して間もない頃の時代に遡る． その頃のコンピュータというと，何千何万のという数の真空管が接続された，会議室一個分くらいのサイズのマシンであった． そのような高価でメンテが大変な機材であるから，当然みんなでシェアして使うことが前提となる． ユーザーがコンピュータにアクセスするため，マシンからは何本かのケーブルが伸び，それぞれにキーボードとスクリーンが接続されていた… これを Terminal とよんでいたのである． 人々は，代わる代わる Terminal の前に座って，計算機との対話を行っていた．

時代は流れ，Windows や Mac などのいわゆるパーソナルコンピュータの出現により，コンピュータはみんなで共有するものではなく，個人が所有するものになった．

最近のクラウドの台頭は，みんなで大きなコンピュータをシェアするという，最初のコンピュータの使われ方に原点回帰していると捉えることもできる． 一方で，スマートフォンやウェアラブルなどのエッジデバイスの普及も盛んであり，個人が複数の"小さな"コンピュータを所有する，という流れも同時に進行しているのである．

+ \ No newline at end of file diff --git a/development/aws/docker-system.html b/development/aws/docker-system.html index 2b6750b9..dfe71ba9 100644 --- a/development/aws/docker-system.html +++ b/development/aws/docker-system.html @@ -5,13 +5,13 @@ Docker 入門 | Toshiki's Note - + - + @@ -97,9 +97,9 @@ RUN rm -rf /opt/* # copy hands-on source code in /root/ -COPY handson/ /root/handson

Dockerfile の中身の説明は詳しくは行わないが，たとえば上のコードで <1> で示したところは， Python 3.7 のインストールを実行している． また， <2> で示したところは， AWS CDK のインストールを行っていることがわかるだろう． このように，リアルな OS で行うのと同じ流れでインストールのコマンドを逐一記述していくことで，自分だけの Docker イメージを作成することができる． 一度イメージを作成すれば，それを配布することで，他者も同一の計算環境を簡単に再構成することができる．

"ぼくの環境ではそのプログラム走ったのにな…" というのは，プログラミング初心者ではよく耳にする会話だが， Docker を使いこなせばそのような心配とは無縁である． そのような意味で，クラウド以外の場面でも， Docker の有用性・汎用性は極めて高い．

コンテナを用いた仮想計算環境ツールとして Docker を紹介したが， ほかに選択肢はないのか？ よくぞ聞いてくれた！ Docker の登場以降，複数のコンテナベースの仮想環境ツールが開発されてきた． いずれのツールも，概念や API については Docker と共通するものが多いが，Docker にはない独自の特徴を提供している． ここではその中でも有名ないくつかを紹介しよう．

Singularity は科学計算や HPC (High Performance Computing) の分野で人気の高いコンテナプラットフォームである． Singularity では大学・研究機関の HPC クラスターでの運用に適したような設計が施されている． たとえば， Docker は基本的には root 権限で実行されるのに対し， Singularity はユーザー権限 (コマンドを実行したユーザー自身) でプログラムが実行される． root 権限での実行は Web サーバーのように個人・企業がある特定のサービスのために運用するサーバーでは問題ないが，多数のユーザーが多様な目的で計算を実行する HPC クラスターでは問題となる． また，Singularity は独自のイメージの作成方法・エコシステムをもっているが， Docker イメージを Singularity のイメージに変換し実行する機能も有している．

podman は Red Hat 社によって開発されたもう一つのコンテナプラットフォームである． podman は基本的に Docker と同一のコマンドを採用しているが，実装は Red Hat によってスクラッチから行われた． podman では， Singularity と同様にユーザー権限でのプログラムの実行を可能であり，クラウドおよび HPC の両方の環境に対応するコンテナプラットフォームを目指して作られた． また，その名前にあるとおり pod とよばれる独自の概念が導入されている．

著者の個人的な意見としては，現時点では Docker をマスターしておけば当面は困ることはないと考えるが，興味のある読者はぜひこれらのツールも試してみてはいかがだろうか？

Elastic Container Service (ECS) ​

ECS のアイコン

ここまでに説明してきたように， Docker を使うことで仮想計算環境を簡単に複製・起動することが可能になる． 本章の最後の話題として， AWS 上で Docker を使った計算システムを構築する方法を解説しよう．

Elastic Container Service (ECS) とは， Docker を使った計算機クラスターを AWS 上に作成するためのツールである (figure_title)． ECS を使用することで， Docker にパッケージされたアプリケーションを計算機クラスターに投入したり，計算機クラスターのインスタンスを追加・削除する操作 (=スケーリング) を行うことができる．

ECS の概要を示したのが figure_title である． ECS は，タスク (Task) と呼ばれる単位で管理された計算ジョブを受け付ける． システムにタスクが投入されると，ECS は最初にタスクで指定された Docker イメージを外部レジストリからダウンロードしてくる． 外部レジストリとしては， Docker Hub や AWS 独自の Docker レジストリである ECR (Elastic Container Registry) を指定することができる．

ECS の次の重要な役割はタスクの配置である． あらかじめ定義されたクラスター内で，計算負荷が小さい仮想インスタンスを選び出し，そこに Docker イメージを配置することで指定された計算タスクが開始される． "計算負荷が小さい仮想インスタンスを選び出す" と言ったが，具体的にどのような戦略・ポリシーでこの選択を行うかは，ユーザーの指定したパラメータに従う．

また，クラスターのスケーリングも ECS における重要な概念である． スケーリングとは，クラスター内のインスタンスの計算負荷をモニタリングし，計算負荷に応じてインスタンスの起動・停止を行う操作を指す． クラスター全体の計算負荷が指定された閾値 (たとえば 80%の稼働率) を超えていた場合，新たな仮想インスタンスをクラスター内に立ち上げる操作を scale-out (スケールアウト) とよび， 負荷が減った場合に不要なインスタンスを停止する操作を scale-in (スケールイン) とよぶ． クラスターのスケーリングは， ECS がほかの AWS のサービスと連携することで実現される． 具体的には， EC2 の Auto scaling group (ASG) や Fargate の２つの選択肢が多くの場合選択される． ASG については ???, Fargate については ??? でより詳細に解説する．

これら一連のタスクの管理を， ECS は自動でやってくれる． クラスターのスケーリングやタスクの配置に関してのパラメータを一度指定してしまえば，ユーザーは (ほとんどなにも考えずに) 大量のタスクを投入することができる． クラスターのスケーリングによってタスクの量にちょうど十分なだけのインスタンスが起動し，タスクが完了した後は不要なインスタンスはすべて停止される．

さて，ここまで説明的な話が続いてしまったが，次章からは早速 Docker と AWS を使って大規模な並列計算システムを構築していこう！

ECS の概要

- +COPY handson/ /root/handson

Dockerfile の中身の説明は詳しくは行わないが，たとえば上のコードで <1> で示したところは， Python 3.7 のインストールを実行している． また， <2> で示したところは， AWS CDK のインストールを行っていることがわかるだろう． このように，リアルな OS で行うのと同じ流れでインストールのコマンドを逐一記述していくことで，自分だけの Docker イメージを作成することができる． 一度イメージを作成すれば，それを配布することで，他者も同一の計算環境を簡単に再構成することができる．

"ぼくの環境ではそのプログラム走ったのにな…" というのは，プログラミング初心者ではよく耳にする会話だが， Docker を使いこなせばそのような心配とは無縁である． そのような意味で，クラウド以外の場面でも， Docker の有用性・汎用性は極めて高い．

コンテナを用いた仮想計算環境ツールとして Docker を紹介したが， ほかに選択肢はないのか？ よくぞ聞いてくれた！ Docker の登場以降，複数のコンテナベースの仮想環境ツールが開発されてきた． いずれのツールも，概念や API については Docker と共通するものが多いが，Docker にはない独自の特徴を提供している． ここではその中でも有名ないくつかを紹介しよう．

Singularity は科学計算や HPC (High Performance Computing) の分野で人気の高いコンテナプラットフォームである． Singularity では大学・研究機関の HPC クラスターでの運用に適したような設計が施されている． たとえば， Docker は基本的には root 権限で実行されるのに対し， Singularity はユーザー権限 (コマンドを実行したユーザー自身) でプログラムが実行される． root 権限での実行は Web サーバーのように個人・企業がある特定のサービスのために運用するサーバーでは問題ないが，多数のユーザーが多様な目的で計算を実行する HPC クラスターでは問題となる． また，Singularity は独自のイメージの作成方法・エコシステムをもっているが， Docker イメージを Singularity のイメージに変換し実行する機能も有している．

podman は Red Hat 社によって開発されたもう一つのコンテナプラットフォームである． podman は基本的に Docker と同一のコマンドを採用しているが，実装は Red Hat によってスクラッチから行われた． podman では， Singularity と同様にユーザー権限でのプログラムの実行を可能であり，クラウドおよび HPC の両方の環境に対応するコンテナプラットフォームを目指して作られた． また，その名前にあるとおり pod とよばれる独自の概念が導入されている．

著者の個人的な意見としては，現時点では Docker をマスターしておけば当面は困ることはないと考えるが，興味のある読者はぜひこれらのツールも試してみてはいかがだろうか？

Elastic Container Service (ECS) ​

ECS のアイコン

ここまでに説明してきたように， Docker を使うことで仮想計算環境を簡単に複製・起動することが可能になる． 本章の最後の話題として， AWS 上で Docker を使った計算システムを構築する方法を解説しよう．

Elastic Container Service (ECS) とは， Docker を使った計算機クラスターを AWS 上に作成するためのツールである (figure_title)． ECS を使用することで， Docker にパッケージされたアプリケーションを計算機クラスターに投入したり，計算機クラスターのインスタンスを追加・削除する操作 (=スケーリング) を行うことができる．

ECS の概要を示したのが figure_title である． ECS は，タスク (Task) と呼ばれる単位で管理された計算ジョブを受け付ける． システムにタスクが投入されると，ECS は最初にタスクで指定された Docker イメージを外部レジストリからダウンロードしてくる． 外部レジストリとしては， Docker Hub や AWS 独自の Docker レジストリである ECR (Elastic Container Registry) を指定することができる．

ECS の次の重要な役割はタスクの配置である． あらかじめ定義されたクラスター内で，計算負荷が小さい仮想インスタンスを選び出し，そこに Docker イメージを配置することで指定された計算タスクが開始される． "計算負荷が小さい仮想インスタンスを選び出す" と言ったが，具体的にどのような戦略・ポリシーでこの選択を行うかは，ユーザーの指定したパラメータに従う．

また，クラスターのスケーリングも ECS における重要な概念である． スケーリングとは，クラスター内のインスタンスの計算負荷をモニタリングし，計算負荷に応じてインスタンスの起動・停止を行う操作を指す． クラスター全体の計算負荷が指定された閾値 (たとえば 80%の稼働率) を超えていた場合，新たな仮想インスタンスをクラスター内に立ち上げる操作を scale-out (スケールアウト) とよび， 負荷が減った場合に不要なインスタンスを停止する操作を scale-in (スケールイン) とよぶ． クラスターのスケーリングは， ECS がほかの AWS のサービスと連携することで実現される． 具体的には， EC2 の Auto scaling group (ASG) や Fargate の２つの選択肢が多くの場合選択される． ASG については ???, Fargate については ??? でより詳細に解説する．

これら一連のタスクの管理を， ECS は自動でやってくれる． クラスターのスケーリングやタスクの配置に関してのパラメータを一度指定してしまえば，ユーザーは (ほとんどなにも考えずに) 大量のタスクを投入することができる． クラスターのスケーリングによってタスクの量にちょうど十分なだけのインスタンスが起動し，タスクが完了した後は不要なインスタンスはすべて停止される．

さて，ここまで説明的な話が続いてしまったが，次章からは早速 Docker と AWS を使って大規模な並列計算システムを構築していこう！

ECS の概要

+ \ No newline at end of file diff --git a/development/aws/handson-bashoutter.html b/development/aws/handson-bashoutter.html index 52485983..aaf9f451 100644 --- a/development/aws/handson-bashoutter.html +++ b/development/aws/handson-bashoutter.html @@ -5,13 +5,13 @@ Hands-on #6: Bashoutter | Toshiki's Note - + - + @@ -441,9 +441,9 @@ "likes": 1.0, ... } -]

最後に， DELETE リクエストを送ることで俳句をデータベースから削除しよう． XXXX は item_id の値で置き換えたうえで次のコマンドを実行する．

shell
$ http DELETE "${ENDPOINT_URL}/haiku/XXXX"
$ http DELETE "${ENDPOINT_URL}/haiku/XXXX"

再び GET リクエストを送ることで，返り値が空 ([]) になっていることを確認しよう．

これで，俳句の投稿・取得・削除そしていいねの追加，といった基本的な API がきちんと動作していることが確認できた．

大量の API リクエストをシミュレートする ​

さて，前節ではマニュアルで一つずつ俳句を投稿した． 多数のユーザーがいるような SNS では，1 秒間に数千件以上の投稿がされている． 今回はサーバーレスアーキテクチャを採用したことで，そのような瞬間的な大量アクセスにも容易に対応できるようなシステムが自動的に構築されている． このポイントを実証するため，ここでは大量の API が送信された状況をシミュレートしてみよう．

handson/bashoutter/client.py に，大量の API リクエストをシミュレートするためのプログラムが書かれている． このプログラムを使用すると， POST /haiku の API リクエストを指定された回数だけ実行することができる．

テストとして， API を 300 回実行してみよう． 次のコマンドを実行する．

shell
$ python client.py $ENDPOINT_URL post_many 300
$ python client.py $ENDPOINT_URL post_many 300

数秒のうちに実行が完了するだろう． これがもし，単一のサーバーからなる API だったとしたら，このような大量のリクエストの処理にはもっと時間がかかっただろう． 最悪の場合には，サーバーダウンにもつながっていたかもしれない． したがって，今回作成したサーバーレスアプリケーションは，とてもシンプルながらも 1 秒間に数百件の処理を行えるような，スケーラブルなクラウドシステムであることがわかる． サーバーレスでクラウドを設計することの利点を垣間見ることができただろうか？

先述のコマンドにより大量の俳句を投稿するとデータベースに無駄なデータがどんどん溜まってしまう． データベースを完全に空にするには，次のコマンドを使用する．

shell
$ python client.py $ENDPOINT_URL clear_database
$ python client.py $ENDPOINT_URL clear_database

Bashoutter GUI を動かしてみる ​

前節ではコマンドラインから API を送信する演習を行った． ウェブアプリケーションでは，これと同じことがウェブブラウザの背後で行われ，ページのコンテンツが表示されている (??? 参照)． 最後に， API が GUI と統合されるとどうなるのか，見てみよう．

CDK のコードで， Public access mode の S3 バケットを作成したことを思い出してほしい． 最初のステップとして，ここにウェブサイトのコンテンツをアップロードしよう． ハンズオンのソースコードの中に gui/dist というフォルダが見つかるはずである． ここにはビルド済みのウェブサイトの静的コンテンツ (HTML/CSS/JavaScript) が入っている． AWS CLI のコマンドを使うことでこれらのファイルを S3 にアップロードしよう．

shell
$ aws s3 cp --recursive ./gui/dist s3://<BUCKET_NAME>
$ aws s3 cp --recursive ./gui/dist s3://<BUCKET_NAME>

コマンドを実行する際は， Bashoutter ハンズオンのディレクトリから行うこと (./gui/dist に注目)，そして <BUCKET_NAME> にはデプロイした自身のバケットの名前が入る点に注意． 念のため，AWS コンソールにログインし，バケットにファイルがアップロードされている点を確認しておこう．

なお，今回は GUI の説明はとくに行わないが， Bashoutter のウェブサイトは Vue.js と Vuetify という UI フレームワークを使って作成した． Vue を使うことで， Single page application (SPA) の技術でウェブサイトの画面がレンダリングされる． ソースコードは handson/bashoutter/gui のディレクトリの中にあるので，興味のある読者は確認してみるとよい．

アップトードが完了したところで，続いてデプロイを実行したときにコマンドラインの出力を見直してみよう． Bashoutter.BucketUrl= で与えられた URL が見つかるはずである (figure_title)． これは，先述したとおり， Public access mode の S3 バケットの URL である．

ウェブブラウザを開き，アドレスバーに S3 の URL を入力しへアクセスしてみよう． すると， figure_title のようなページが表示されるはずである．

"Bashoutter" の GUI 画面

ページが表示されたら，一番上の "API Endpoint URL" と書いてあるテキストボックスに，今回デプロイした API Gateway の URL を入力する (今回のアプリケーションでは，API Gateway の URL はランダムに割り当てられるのでこのような GUI の仕様になっている)． そうしたら，画面の "REFRESH" と書いてあるボタンを押してみよう． データベースに俳句が登録済みであれば，俳句の一覧が表示されるはずである． 各俳句の左下にあるハートのアイコンをクリックすることで， "like" の票を入れることができる．

新しい俳句を投稿するには，五七五と投稿者の名前を入力して， "POST" を押す． "POST" を押した後は，再び "REFRESH" ボタンを押すことで最新の俳句のリストをデータベースから取得する．

アプリケーションの削除 ​

これで， Bashoutter プロジェクトが完成した！ この SNS は，インターネットを通じて世界のどこからでもアクセスできる状態にある． また， 大量の API リクエストをシミュレートする で見たように，大量のユーザーの同時アクセスによる負荷がかかっても，柔軟にスケールが行われ遅延なく処理を行うことができる． 極めて簡素ながらも，立派なウェブサービスとしてのスペックは満たしているのである！

Bashoutter アプリを存分に楽しむことができたら，最後に忘れずにスタックを削除しよう．

コマンドラインからスタックの削除を実行するには，次のコマンドを使う．

shell
$ cdk destroy
$ cdk destroy

CDK のバージョンによっては S3 のバケットが空でないと， cdk destroy がエラーを出力する場合がある． この場合はスタックを削除する前に， S3 バケットの中身をすべて削除しなければならない．

コンソールから実行するには， S3 コンソールに行き，バケットの中身を開いたうえで，すべてのファイルを選択し， "Actions" → "Delete" を実行すればよいい．

コマンドラインから実行するには， 次のコマンドを使う． <BUCKET NAME> のところは，自分の バケットの名前 ("BashoutterBucketXXXX" というパターンの名前がついているはずである) に置き換えることを忘れずに．

shell
$ aws s3 rm <BUCKET NAME> --recursive
$ aws s3 rm <BUCKET NAME> --recursive

小括 ​

ここまでが，本書第三部の内容であった．

第三部では，クラウドの応用として，一般の人に使ってもらうようなウェブアプリケーション・データベースをどのようにして作るのか，という点に焦点を当てて，説明を行った． その中で，従来的なクラウドシステムの設計と，ここ数年の最新の設計方法であるサーバーレスアーキテクチャについて解説した． ??? では， AWS でのサーバーレスの実践として， Lambda, S3, DynamoDB のハンズオンを行った． 最後に， Hands-on #6: Bashoutter では，これらの技術を統合することで，完全サーバーレスなウェブアプリケーション "Bashoutter" を作成した．

これらの演習を通じて，世の中のウェブサービスがどのようにしてでき上がっているのか，少し理解が深まっただろうか？ また，そのようなウェブアプリケーションを自分が作りたいと思ったとき，今回のハンズオンがその出発点となることができたならば幸いである．

- +]

最後に， DELETE リクエストを送ることで俳句をデータベースから削除しよう． XXXX は item_id の値で置き換えたうえで次のコマンドを実行する．

shell
$ http DELETE "${ENDPOINT_URL}/haiku/XXXX"
$ http DELETE "${ENDPOINT_URL}/haiku/XXXX"

再び GET リクエストを送ることで，返り値が空 ([]) になっていることを確認しよう．

これで，俳句の投稿・取得・削除そしていいねの追加，といった基本的な API がきちんと動作していることが確認できた．

大量の API リクエストをシミュレートする ​

さて，前節ではマニュアルで一つずつ俳句を投稿した． 多数のユーザーがいるような SNS では，1 秒間に数千件以上の投稿がされている． 今回はサーバーレスアーキテクチャを採用したことで，そのような瞬間的な大量アクセスにも容易に対応できるようなシステムが自動的に構築されている． このポイントを実証するため，ここでは大量の API が送信された状況をシミュレートしてみよう．

handson/bashoutter/client.py に，大量の API リクエストをシミュレートするためのプログラムが書かれている． このプログラムを使用すると， POST /haiku の API リクエストを指定された回数だけ実行することができる．

テストとして， API を 300 回実行してみよう． 次のコマンドを実行する．

shell
$ python client.py $ENDPOINT_URL post_many 300
$ python client.py $ENDPOINT_URL post_many 300

数秒のうちに実行が完了するだろう． これがもし，単一のサーバーからなる API だったとしたら，このような大量のリクエストの処理にはもっと時間がかかっただろう． 最悪の場合には，サーバーダウンにもつながっていたかもしれない． したがって，今回作成したサーバーレスアプリケーションは，とてもシンプルながらも 1 秒間に数百件の処理を行えるような，スケーラブルなクラウドシステムであることがわかる． サーバーレスでクラウドを設計することの利点を垣間見ることができただろうか？

先述のコマンドにより大量の俳句を投稿するとデータベースに無駄なデータがどんどん溜まってしまう． データベースを完全に空にするには，次のコマンドを使用する．

shell
$ python client.py $ENDPOINT_URL clear_database
$ python client.py $ENDPOINT_URL clear_database

Bashoutter GUI を動かしてみる ​

前節ではコマンドラインから API を送信する演習を行った． ウェブアプリケーションでは，これと同じことがウェブブラウザの背後で行われ，ページのコンテンツが表示されている (??? 参照)． 最後に， API が GUI と統合されるとどうなるのか，見てみよう．

CDK のコードで， Public access mode の S3 バケットを作成したことを思い出してほしい． 最初のステップとして，ここにウェブサイトのコンテンツをアップロードしよう． ハンズオンのソースコードの中に gui/dist というフォルダが見つかるはずである． ここにはビルド済みのウェブサイトの静的コンテンツ (HTML/CSS/JavaScript) が入っている． AWS CLI のコマンドを使うことでこれらのファイルを S3 にアップロードしよう．

shell
$ aws s3 cp --recursive ./gui/dist s3://<BUCKET_NAME>
$ aws s3 cp --recursive ./gui/dist s3://<BUCKET_NAME>

コマンドを実行する際は， Bashoutter ハンズオンのディレクトリから行うこと (./gui/dist に注目)，そして <BUCKET_NAME> にはデプロイした自身のバケットの名前が入る点に注意． 念のため，AWS コンソールにログインし，バケットにファイルがアップロードされている点を確認しておこう．

なお，今回は GUI の説明はとくに行わないが， Bashoutter のウェブサイトは Vue.js と Vuetify という UI フレームワークを使って作成した． Vue を使うことで， Single page application (SPA) の技術でウェブサイトの画面がレンダリングされる． ソースコードは handson/bashoutter/gui のディレクトリの中にあるので，興味のある読者は確認してみるとよい．

アップトードが完了したところで，続いてデプロイを実行したときにコマンドラインの出力を見直してみよう． Bashoutter.BucketUrl= で与えられた URL が見つかるはずである (figure_title)． これは，先述したとおり， Public access mode の S3 バケットの URL である．

ウェブブラウザを開き，アドレスバーに S3 の URL を入力しへアクセスしてみよう． すると， figure_title のようなページが表示されるはずである．

"Bashoutter" の GUI 画面

ページが表示されたら，一番上の "API Endpoint URL" と書いてあるテキストボックスに，今回デプロイした API Gateway の URL を入力する (今回のアプリケーションでは，API Gateway の URL はランダムに割り当てられるのでこのような GUI の仕様になっている)． そうしたら，画面の "REFRESH" と書いてあるボタンを押してみよう． データベースに俳句が登録済みであれば，俳句の一覧が表示されるはずである． 各俳句の左下にあるハートのアイコンをクリックすることで， "like" の票を入れることができる．

新しい俳句を投稿するには，五七五と投稿者の名前を入力して， "POST" を押す． "POST" を押した後は，再び "REFRESH" ボタンを押すことで最新の俳句のリストをデータベースから取得する．

アプリケーションの削除 ​

これで， Bashoutter プロジェクトが完成した！ この SNS は，インターネットを通じて世界のどこからでもアクセスできる状態にある． また， 大量の API リクエストをシミュレートする で見たように，大量のユーザーの同時アクセスによる負荷がかかっても，柔軟にスケールが行われ遅延なく処理を行うことができる． 極めて簡素ながらも，立派なウェブサービスとしてのスペックは満たしているのである！

Bashoutter アプリを存分に楽しむことができたら，最後に忘れずにスタックを削除しよう．

コマンドラインからスタックの削除を実行するには，次のコマンドを使う．

shell
$ cdk destroy
$ cdk destroy

CDK のバージョンによっては S3 のバケットが空でないと， cdk destroy がエラーを出力する場合がある． この場合はスタックを削除する前に， S3 バケットの中身をすべて削除しなければならない．

コンソールから実行するには， S3 コンソールに行き，バケットの中身を開いたうえで，すべてのファイルを選択し， "Actions" → "Delete" を実行すればよいい．

コマンドラインから実行するには， 次のコマンドを使う． <BUCKET NAME> のところは，自分の バケットの名前 ("BashoutterBucketXXXX" というパターンの名前がついているはずである) に置き換えることを忘れずに．

shell
$ aws s3 rm <BUCKET NAME> --recursive
$ aws s3 rm <BUCKET NAME> --recursive

小括 ​

ここまでが，本書第三部の内容であった．

第三部では，クラウドの応用として，一般の人に使ってもらうようなウェブアプリケーション・データベースをどのようにして作るのか，という点に焦点を当てて，説明を行った． その中で，従来的なクラウドシステムの設計と，ここ数年の最新の設計方法であるサーバーレスアーキテクチャについて解説した． ??? では， AWS でのサーバーレスの実践として， Lambda, S3, DynamoDB のハンズオンを行った． 最後に， Hands-on #6: Bashoutter では，これらの技術を統合することで，完全サーバーレスなウェブアプリケーション "Bashoutter" を作成した．

これらの演習を通じて，世の中のウェブサービスがどのようにしてでき上がっているのか，少し理解が深まっただろうか？ また，そのようなウェブアプリケーションを自分が作りたいと思ったとき，今回のハンズオンがその出発点となることができたならば幸いである．

+ \ No newline at end of file diff --git a/development/aws/handson-ec2.html b/development/aws/handson-ec2.html index a111708a..1a2dbb38 100644 --- a/development/aws/handson-ec2.html +++ b/development/aws/handson-ec2.html @@ -5,14 +5,14 @@ Hands-on #1: 初めての EC2 インスタンスを起動する | Toshiki's Note - + - + @@ -226,9 +226,9 @@ Python 3.6.10 (default, Feb 10 2020, 19:55:14) [GCC 4.8.5 20150623 (Red Hat 4.8.5-28)] on linux Type "help", "copyright", "credits" or "license" for more information. ->>>

Python のインタープリタが起動した！ Ctrl + D あるいは exit() と入力することで，インタープリタを閉じることができる．

さて，サーバーでのお遊びはこんなところにしておこう (興味があれば各自いろいろと試してみると良い) ． 次のコマンドでログアウトする．

shell
$ exit
$ exit

AWS コンソールから確認 ​

これまでは，すべてコマンドラインから EC2 に関連する操作を行ってきた． EC2 インスタンスの状態を確認したり，サーバーをシャットダウンするなどの操作は，AWS コンソールから実行することもできる． 軽くこれを紹介しよう．

まず，ウェブブラウザを開いて AWS コンソールにログインする． ログインしたら， Services から EC2 を検索(選択)する． 次に，左のサイドバーの Instances とページをたどる. すると， figure_title のような画面が得られるはずである． この画面で，自分のアカウントの管理下にあるインスタンスを確認できる． 同様に，VPC・SG についてもコンソールから確認できる．

EC2 コンソール画面

コンソール右上で，正しいリージョン (今回の場合は ap-northeast-1, Tokyo) が選択されているか，注意する！

前章で CloudFormation について触れたが，今回デプロイしたアプリケーションも，CloudFormation のスタックとして管理されている． スタック (stack) とは， AWS リソースの集合のことを指す． 今回の場合は， VPC/EC2/SG などがスタックの中に含まれている． コンソールで CloudFormation のページに行ってみよう (figure_title)．

CloudFormation コンソール画面

"MyFirstEc2" という名前のスタックがあることが確認できる． クリックをして中身を見てみると，EC2, VPC などのリソースがこのスタックに紐付いていることがわかる．

スタックを削除 ​

これにて，第一回のハンズオンで説明すべき事柄はすべて完了した． 最後に，使わなくなったスタックを削除しよう． スタックの削除には，二つの方法がある．

一つ目の方法は，前節の Cloudformation のコンソール画面で， "Delete" ボタンを押すことである (figure_title)． すると，スタックの状態が "DELETE_IN_PROGRESS" に変わり，削除が完了すると CloudFormation のスタックの一覧から消える．

CloudFormationコンソール画面から，スタックを削除

二つ目の方法は，コマンドラインから行う方法である． 先ほど，デプロイを行ったコマンドラインに戻ろう． そうしたら，次のコマンドを実行する．

shell
$ cdk destroy
$ cdk destroy

このコマンドを実行すると，スタックの削除が始まる． 削除した後は，VPC, EC2 など，すべて跡形もなく消え去っていることを自身で確かめよう． CloudFormation を用いることで関連するすべての AWS リソースを一度に管理・削除することができるので，大変便利である．

スタックの削除は各自で必ず行うこと！ 行わなかった場合， EC2 インスタンスの料金が発生し続けることになる！

また，本ハンズオンのために作成した SSH 鍵ペアも不要なので，削除しておく． まず， EC2 側に登録してある公開鍵を削除する． これも，コンソールおよびコマンドラインの二つの方法で実行できる．

コンソールから実行するには， EC2 の画面に行き，左のサイドバーの Key Pairs を選択する． 鍵の一覧が表示されるので， HirakeGoma とある鍵にチェックを入れ，画面右上の Actions から， Delete を実行する (figure_title)．

EC2でSSH鍵ペアを削除

コマンドラインから実行するには，次のコマンドを使う．

shell
$ aws ec2 delete-key-pair --key-name "HirakeGoma"
$ aws ec2 delete-key-pair --key-name "HirakeGoma"

最後に，ローカルのコンピュータから鍵を削除する．

shell
$ rm -f ~/.ssh/HirakeGoma.pem
$ rm -f ~/.ssh/HirakeGoma.pem

これで，クラウドの片付けもすべて終了だ．

なお，頻繁に EC2 インスタンスを起動したりする場合は，いちいち SSH 鍵を削除する必要はない．

小括 ​

ここまでが，本書の第一部の内容である． 盛りだくさんの内容であったが，ついてこれたであろうか？

??? では，クラウドの定義と用語の説明を行ったあと，なぜクラウドを使うのか，という点を議論した． 続いて ??? では，クラウドを学ぶ具体的なプラットフォームとして AWS を取り上げ， AWS を使用するにあたり最低限必要な知識と用語の説明を行った． さらに， Hands-on #1: 初めての EC2 インスタンスを起動する のハンズオンでは AWS CLI と AWS CDK を使って，自身のプライベートなサーバーを AWS 上に立ち上げる演習を行った．

これらを通じて，いかに簡単に (たった数行のコマンドで！) 仮想サーバーを立ち上げたり，削除したりすることができるか，体験できただろう． 筆者は，??? でクラウドの最も重要な側面はダイナミックに計算リソースを拡大・縮小できることである，と述べた． この言葉の意味が，ハンズオンを通じてより明らかになっただろうか？ ここで学んだ技術を少し応用するだけで，自分のウェブページをホストする仮想サーバーを作成したり，大量のコアを搭載した EC2 インスタンスを用意して科学計算を実行するなど，いろいろなアプリケーションが実現できる．

次章からは，今回学んだクラウドの技術を基に，より現実に即した問題を解くことを体験してもらう． お楽しみに！

- +>>>

Python のインタープリタが起動した！ Ctrl + D あるいは exit() と入力することで，インタープリタを閉じることができる．

さて，サーバーでのお遊びはこんなところにしておこう (興味があれば各自いろいろと試してみると良い) ． 次のコマンドでログアウトする．

shell
$ exit
$ exit

AWS コンソールから確認 ​

これまでは，すべてコマンドラインから EC2 に関連する操作を行ってきた． EC2 インスタンスの状態を確認したり，サーバーをシャットダウンするなどの操作は，AWS コンソールから実行することもできる． 軽くこれを紹介しよう．

まず，ウェブブラウザを開いて AWS コンソールにログインする． ログインしたら， Services から EC2 を検索(選択)する． 次に，左のサイドバーの Instances とページをたどる. すると， figure_title のような画面が得られるはずである． この画面で，自分のアカウントの管理下にあるインスタンスを確認できる． 同様に，VPC・SG についてもコンソールから確認できる．

EC2 コンソール画面

コンソール右上で，正しいリージョン (今回の場合は ap-northeast-1, Tokyo) が選択されているか，注意する！

前章で CloudFormation について触れたが，今回デプロイしたアプリケーションも，CloudFormation のスタックとして管理されている． スタック (stack) とは， AWS リソースの集合のことを指す． 今回の場合は， VPC/EC2/SG などがスタックの中に含まれている． コンソールで CloudFormation のページに行ってみよう (figure_title)．

CloudFormation コンソール画面

"MyFirstEc2" という名前のスタックがあることが確認できる． クリックをして中身を見てみると，EC2, VPC などのリソースがこのスタックに紐付いていることがわかる．

スタックを削除 ​

これにて，第一回のハンズオンで説明すべき事柄はすべて完了した． 最後に，使わなくなったスタックを削除しよう． スタックの削除には，二つの方法がある．

一つ目の方法は，前節の Cloudformation のコンソール画面で， "Delete" ボタンを押すことである (figure_title)． すると，スタックの状態が "DELETE_IN_PROGRESS" に変わり，削除が完了すると CloudFormation のスタックの一覧から消える．

CloudFormationコンソール画面から，スタックを削除

二つ目の方法は，コマンドラインから行う方法である． 先ほど，デプロイを行ったコマンドラインに戻ろう． そうしたら，次のコマンドを実行する．

shell
$ cdk destroy
$ cdk destroy

このコマンドを実行すると，スタックの削除が始まる． 削除した後は，VPC, EC2 など，すべて跡形もなく消え去っていることを自身で確かめよう． CloudFormation を用いることで関連するすべての AWS リソースを一度に管理・削除することができるので，大変便利である．

スタックの削除は各自で必ず行うこと！ 行わなかった場合， EC2 インスタンスの料金が発生し続けることになる！

また，本ハンズオンのために作成した SSH 鍵ペアも不要なので，削除しておく． まず， EC2 側に登録してある公開鍵を削除する． これも，コンソールおよびコマンドラインの二つの方法で実行できる．

コンソールから実行するには， EC2 の画面に行き，左のサイドバーの Key Pairs を選択する． 鍵の一覧が表示されるので， HirakeGoma とある鍵にチェックを入れ，画面右上の Actions から， Delete を実行する (figure_title)．

EC2でSSH鍵ペアを削除

コマンドラインから実行するには，次のコマンドを使う．

shell
$ aws ec2 delete-key-pair --key-name "HirakeGoma"
$ aws ec2 delete-key-pair --key-name "HirakeGoma"

最後に，ローカルのコンピュータから鍵を削除する．

shell
$ rm -f ~/.ssh/HirakeGoma.pem
$ rm -f ~/.ssh/HirakeGoma.pem

これで，クラウドの片付けもすべて終了だ．

なお，頻繁に EC2 インスタンスを起動したりする場合は，いちいち SSH 鍵を削除する必要はない．

小括 ​

ここまでが，本書の第一部の内容である． 盛りだくさんの内容であったが，ついてこれたであろうか？

??? では，クラウドの定義と用語の説明を行ったあと，なぜクラウドを使うのか，という点を議論した． 続いて ??? では，クラウドを学ぶ具体的なプラットフォームとして AWS を取り上げ， AWS を使用するにあたり最低限必要な知識と用語の説明を行った． さらに， Hands-on #1: 初めての EC2 インスタンスを起動する のハンズオンでは AWS CLI と AWS CDK を使って，自身のプライベートなサーバーを AWS 上に立ち上げる演習を行った．

これらを通じて，いかに簡単に (たった数行のコマンドで！) 仮想サーバーを立ち上げたり，削除したりすることができるか，体験できただろう． 筆者は，??? でクラウドの最も重要な側面はダイナミックに計算リソースを拡大・縮小できることである，と述べた． この言葉の意味が，ハンズオンを通じてより明らかになっただろうか？ ここで学んだ技術を少し応用するだけで，自分のウェブページをホストする仮想サーバーを作成したり，大量のコアを搭載した EC2 インスタンスを用意して科学計算を実行するなど，いろいろなアプリケーションが実現できる．

次章からは，今回学んだクラウドの技術を基に，より現実に即した問題を解くことを体験してもらう． お楽しみに！

+ \ No newline at end of file diff --git a/development/aws/handson-jupyter.html b/development/aws/handson-jupyter.html index bd78593e..e1df13e5 100644 --- a/development/aws/handson-jupyter.html +++ b/development/aws/handson-jupyter.html @@ -5,13 +5,13 @@ Hands-on #2: AWS でディープラーニングを実践 | Toshiki's Note - + - + @@ -279,9 +279,9 @@ plt.title("Prediction: {}".format(output.data.max(1, keepdim=True)[1][i].item())) plt.xticks([]) plt.yticks([]) -plt.show()

学習した CNN による，MNIST画像の推論結果

最後に，学習したニューラルネットワークのパラメータを mnist_cnn.pt というファイル名で保存しておこう． これで，将来いつでも今回学習したモデルを再現し，別の実験に使用することができる．

python
torch.save(model.state_dict(), "mnist_cnn.pt")
torch.save(model.state_dict(), "mnist_cnn.pt")

以上が， AWS クラウドの仮想サーバーを立ち上げ，最初のディープラーニングの計算を行う一連の流れである． MNIST 文字認識のタスクを行うニューラルネットを，クラウド上の GPU を使って高速に学習させ，現実的な問題を一つ解くことができたのである． 興味のある読者は，今回のハンズオンを雛形に，自分の所望の計算を走らせてみるとよいだろう．

スタックの削除 ​

これにて，ハンズオン第二回の内容はすべて説明した． クラウドの利用料金を最小化するため，使い終わった EC2 インスタンスはすぐさま削除しよう．

ハンズオン第一回と同様に， AWS の CloudFormation コンソールか， AWS CLI により削除を実行する (詳細は ??? 参照)．

shell
$ cdk destroy
$ cdk destroy

スタックの削除は各自で必ず行うこと！ 行わなかった場合，EC2 インスタンスの料金が発生し続けることになる！ g4dn.xlarge は $0.71 / hour の料金設定なので，一日起動しつづけると約$17 の請求が発生することになる！

AWS のバジェットアラート

AWS の初心者が (あるいは経験者も) しばしば陥る失敗が，インスタンスの停止忘れなどで無駄なリソースがクラウドで放置されてしまい，巨大な額の請求が届く，というミスだ． 特に，開発を行っている間はこのような事態は起こりうるものだと思って，備えておかなければならない． このような事態を未然に防ぐため， AWS Budgets という機能が無料で提供されている． AWS Budgets を利用することで，月の利用金額がある閾値を超えた場合にユーザーにメールが送信される，などのアラートを設定することができる． 詳細な手順は AWS の公式ブログ "Getting Started with AWS Budgets" を参照のこと． 本書の読者も，ぜひこのタイミングでアラートを設定しておくことを推奨する．

- +plt.show()

学習した CNN による，MNIST画像の推論結果

最後に，学習したニューラルネットワークのパラメータを mnist_cnn.pt というファイル名で保存しておこう． これで，将来いつでも今回学習したモデルを再現し，別の実験に使用することができる．

python
torch.save(model.state_dict(), "mnist_cnn.pt")
torch.save(model.state_dict(), "mnist_cnn.pt")

以上が， AWS クラウドの仮想サーバーを立ち上げ，最初のディープラーニングの計算を行う一連の流れである． MNIST 文字認識のタスクを行うニューラルネットを，クラウド上の GPU を使って高速に学習させ，現実的な問題を一つ解くことができたのである． 興味のある読者は，今回のハンズオンを雛形に，自分の所望の計算を走らせてみるとよいだろう．

スタックの削除 ​

これにて，ハンズオン第二回の内容はすべて説明した． クラウドの利用料金を最小化するため，使い終わった EC2 インスタンスはすぐさま削除しよう．

ハンズオン第一回と同様に， AWS の CloudFormation コンソールか， AWS CLI により削除を実行する (詳細は ??? 参照)．

shell
$ cdk destroy
$ cdk destroy

スタックの削除は各自で必ず行うこと！ 行わなかった場合，EC2 インスタンスの料金が発生し続けることになる！ g4dn.xlarge は $0.71 / hour の料金設定なので，一日起動しつづけると約$17 の請求が発生することになる！

AWS のバジェットアラート

AWS の初心者が (あるいは経験者も) しばしば陥る失敗が，インスタンスの停止忘れなどで無駄なリソースがクラウドで放置されてしまい，巨大な額の請求が届く，というミスだ． 特に，開発を行っている間はこのような事態は起こりうるものだと思って，備えておかなければならない． このような事態を未然に防ぐため， AWS Budgets という機能が無料で提供されている． AWS Budgets を利用することで，月の利用金額がある閾値を超えた場合にユーザーにメールが送信される，などのアラートを設定することができる． 詳細な手順は AWS の公式ブログ "Getting Started with AWS Budgets" を参照のこと． 本書の読者も，ぜひこのタイミングでアラートを設定しておくことを推奨する．

+ \ No newline at end of file diff --git a/development/aws/handson-qabot.html b/development/aws/handson-qabot.html index b4e519e8..92880714 100644 --- a/development/aws/handson-qabot.html +++ b/development/aws/handson-qabot.html @@ -5,13 +5,13 @@ Hands-on #3: AWS で自動質問回答ボットを走らせる | Toshiki's Note - + - + @@ -194,9 +194,9 @@ $ pip install -r requirements.txt # デプロイを実行 -$ cdk deploy

デプロイのコマンドが無事に実行されれば， figure_title のような出力が得られるはずである．

CDKデプロイ実行後の出力

AWS コンソールにログインして，デプロイされたスタックの中身を確認してみよう． コンソールから，ECS のページに行くと figure_title のような画面が表示されるはずである． EcsClusterQaBot-XXXX という名前ついたクラスターを見つけよう．

Cluster というのが，先ほど説明したとおり，複数の仮想インスタンスを束ねる一つの単位である． figure_title で， FARGATE という文字の下に 0 Running tasks, 0 Pending tasks と表示されていることを確認しよう． この時点では一つもタスクが走っていないので，数字はすべて 0 になっている．

ECS コンソール画面

続いて，この画面の左のメニューバーから Task Definitions という項目を見つけ，クリックしよう． 移動した先のページで EcsClusterQaBotEcsClusterQaBotTaskDefXXXX という項目が見つかるので，開く． 開いた先のページをスクロールすると figure_title に示したような情報が見つかるだろう． 使用する CPU ・メモリーの量や， Docker container の実行に関する設定などが，この Task Definition の画面から確認することができる．

Task definition の確認

タスクの実行 ​

それでは，質問をデプロイしたクラウドに提出してみよう．

ECS にタスクを投入するのはやや複雑なので，タスクの投入を簡単にするプログラム (run_task.py) を用意した (handson/qa-bot/run_task.py)．

次のようなコマンドで，ECS クラスターに新しい質問を投入することができる．

shell
$ python run_task.py ask "A giant peach was flowing in the river. She picked it up and brought it home. Later, a healthy baby was born from the peach. She named the baby Momotaro." "What is the name of the baby?"
$ python run_task.py ask "A giant peach was flowing in the river. She picked it up and brought it home. Later, a healthy baby was born from the peach. She named the baby Momotaro." "What is the name of the baby?"

run_task.py を実行するには， コマンドラインで AWS の認証情報が設定されていることが前提である．

"ask" の引数に続き，文脈 (context) と質問 (question) を引数として渡している．

このコマンドを実行すると， "Waiting for the task to finish…" と出力が表示され，回答を得るまでしばらく待たされる． この間， AWS では ECS がタスクを受理し，新しい Fargate のインスタンスを起動し， Docker イメージをそのインスタンスに配置する，という一連の処理がなされている． AWS コンソールから，この一連の様子をモニタリングしてみよう．

先ほどの ECS コンソール画面にもどり，クラスターの名前をクリックすることで，クラスターの詳細画面を開く． 次に， "Tasks" という名前のタブがあるので，それを開く (figure_title)． すると，実行中のタスクの一覧が表示されるだろう．

ECS のタスクの実行状況をモニタリング

figure_title で見て取れるように， "Last status = Pending" となっていることから，この時点では，タスクを実行する準備をしている段階である，ということがわかる． Fargate のインスタンスを起動し， Docker image を配置するまでおよそ 1-2 分の時間がかかる．

しばらく待つうちに， Status が "RUNNING" に遷移し，計算が始まる． 計算が終わると， Status は "STOPPED" に遷移し， ECS によって Fargate インスタンスは自動的にシャットダウンされる．

figure_title の画面から， "Task" の列にあるタスク ID クリックすることで，タスクの詳細画面を開いてみよう (figure_title)． "Last status", "Platform version" など，タスクの情報が表示されている． また， "Logs" のタブを開くことで，コンテナの吐き出した実行ログを閲覧することができる．

質問タスクの実行結果

さて， run_task.py を実行したコマンドラインに戻ってきてみると， figure_title のような出力が得られているはずである． "Momotaro" という正しい回答が返ってきている！

質問タスクの実行結果

タスクの同時実行 ​

さて，先ほどはたった一つの質問を投入したわけだが，今回設計したアプリケーションは， ECS と Fargate を使うことで同時にたくさんの質問を処理することができる． 実際に，たくさんの質問を一度に投入してみよう． run_task.py に ask_many というオプションを付けることで，複数の質問を一度に送信できる． 質問の内容は handson/qa-bot/problems.json に定義されている．

次のようなコマンドを実行しよう．

shell
$ python run_task.py ask_many
$ python run_task.py ask_many

このコマンドを実行した後で，先ほどの ECS コンソールに行き，タスクの一覧を見てみよう (figure_title)． 複数の Fargate インスタンスが起動され，タスクが並列に実行されているのがわかる．

複数の質問タスクを同時に投入する

すべてのタスクのステータスが "STOPPED" になったことを確認した上で，質問への回答を取得しよう． それには，次のコマンドを実行する．

shell
$ python run_task.py list_answers
$ python run_task.py list_answers

結果として， figure_title のような出力が得られるだろう． 複雑な文章問題に対し，高い正答率で回答できていることがわかるだろう．

の実行結果

おめでとう！ ここまでついてこれた読者はとても初歩的ながらも，深層学習による言語モデルを使って自動で質問への回答を生成するシステムを創り上げることができた！ それも，数百の質問にも同時に対応できるような，とても高いスケーラビリティーをもったシステムである！ 今回は GUI (Graphical User Interface) を用意することはしなかったが，このシステムに簡単な GUI を追加すればなかなか立派なウェブサービスとして運用できるだろう．

run_task.py で質問を投入し続けると，回答を記録しているデータベースにどんどんエントリーが溜まっていく． これらのエントリーをすべて消去するには，次のコマンドを使う．

shell
$ python run_task.py clear
$ python run_task.py clear

スタックの削除 ​

これにて，今回のハンズオンは終了である． 最後にスタックを削除しよう．

スタックを削除するには，前回までと同様に， AWS コンソールにログインし CloudFormation の画面から DELETE ボタンをクリックするか，コマンドラインからコマンドを実行する． コマンドラインから行う場合は，次のコマンドを使用する．

shell
$ cdk destroy
$ cdk destroy
- +$ cdk deploy

デプロイのコマンドが無事に実行されれば， figure_title のような出力が得られるはずである．

CDKデプロイ実行後の出力

AWS コンソールにログインして，デプロイされたスタックの中身を確認してみよう． コンソールから，ECS のページに行くと figure_title のような画面が表示されるはずである． EcsClusterQaBot-XXXX という名前ついたクラスターを見つけよう．

Cluster というのが，先ほど説明したとおり，複数の仮想インスタンスを束ねる一つの単位である． figure_title で， FARGATE という文字の下に 0 Running tasks, 0 Pending tasks と表示されていることを確認しよう． この時点では一つもタスクが走っていないので，数字はすべて 0 になっている．

ECS コンソール画面

続いて，この画面の左のメニューバーから Task Definitions という項目を見つけ，クリックしよう． 移動した先のページで EcsClusterQaBotEcsClusterQaBotTaskDefXXXX という項目が見つかるので，開く． 開いた先のページをスクロールすると figure_title に示したような情報が見つかるだろう． 使用する CPU ・メモリーの量や， Docker container の実行に関する設定などが，この Task Definition の画面から確認することができる．

Task definition の確認

タスクの実行 ​

それでは，質問をデプロイしたクラウドに提出してみよう．

ECS にタスクを投入するのはやや複雑なので，タスクの投入を簡単にするプログラム (run_task.py) を用意した (handson/qa-bot/run_task.py)．

次のようなコマンドで，ECS クラスターに新しい質問を投入することができる．

shell
$ python run_task.py ask "A giant peach was flowing in the river. She picked it up and brought it home. Later, a healthy baby was born from the peach. She named the baby Momotaro." "What is the name of the baby?"
$ python run_task.py ask "A giant peach was flowing in the river. She picked it up and brought it home. Later, a healthy baby was born from the peach. She named the baby Momotaro." "What is the name of the baby?"

run_task.py を実行するには， コマンドラインで AWS の認証情報が設定されていることが前提である．

"ask" の引数に続き，文脈 (context) と質問 (question) を引数として渡している．

このコマンドを実行すると， "Waiting for the task to finish…" と出力が表示され，回答を得るまでしばらく待たされる． この間， AWS では ECS がタスクを受理し，新しい Fargate のインスタンスを起動し， Docker イメージをそのインスタンスに配置する，という一連の処理がなされている． AWS コンソールから，この一連の様子をモニタリングしてみよう．

先ほどの ECS コンソール画面にもどり，クラスターの名前をクリックすることで，クラスターの詳細画面を開く． 次に， "Tasks" という名前のタブがあるので，それを開く (figure_title)． すると，実行中のタスクの一覧が表示されるだろう．

ECS のタスクの実行状況をモニタリング

figure_title で見て取れるように， "Last status = Pending" となっていることから，この時点では，タスクを実行する準備をしている段階である，ということがわかる． Fargate のインスタンスを起動し， Docker image を配置するまでおよそ 1-2 分の時間がかかる．

しばらく待つうちに， Status が "RUNNING" に遷移し，計算が始まる． 計算が終わると， Status は "STOPPED" に遷移し， ECS によって Fargate インスタンスは自動的にシャットダウンされる．

figure_title の画面から， "Task" の列にあるタスク ID クリックすることで，タスクの詳細画面を開いてみよう (figure_title)． "Last status", "Platform version" など，タスクの情報が表示されている． また， "Logs" のタブを開くことで，コンテナの吐き出した実行ログを閲覧することができる．

質問タスクの実行結果

さて， run_task.py を実行したコマンドラインに戻ってきてみると， figure_title のような出力が得られているはずである． "Momotaro" という正しい回答が返ってきている！

質問タスクの実行結果

タスクの同時実行 ​

さて，先ほどはたった一つの質問を投入したわけだが，今回設計したアプリケーションは， ECS と Fargate を使うことで同時にたくさんの質問を処理することができる． 実際に，たくさんの質問を一度に投入してみよう． run_task.py に ask_many というオプションを付けることで，複数の質問を一度に送信できる． 質問の内容は handson/qa-bot/problems.json に定義されている．

次のようなコマンドを実行しよう．

shell
$ python run_task.py ask_many
$ python run_task.py ask_many

このコマンドを実行した後で，先ほどの ECS コンソールに行き，タスクの一覧を見てみよう (figure_title)． 複数の Fargate インスタンスが起動され，タスクが並列に実行されているのがわかる．

複数の質問タスクを同時に投入する

すべてのタスクのステータスが "STOPPED" になったことを確認した上で，質問への回答を取得しよう． それには，次のコマンドを実行する．

shell
$ python run_task.py list_answers
$ python run_task.py list_answers

結果として， figure_title のような出力が得られるだろう． 複雑な文章問題に対し，高い正答率で回答できていることがわかるだろう．

の実行結果

おめでとう！ ここまでついてこれた読者はとても初歩的ながらも，深層学習による言語モデルを使って自動で質問への回答を生成するシステムを創り上げることができた！ それも，数百の質問にも同時に対応できるような，とても高いスケーラビリティーをもったシステムである！ 今回は GUI (Graphical User Interface) を用意することはしなかったが，このシステムに簡単な GUI を追加すればなかなか立派なウェブサービスとして運用できるだろう．

run_task.py で質問を投入し続けると，回答を記録しているデータベースにどんどんエントリーが溜まっていく． これらのエントリーをすべて消去するには，次のコマンドを使う．

shell
$ python run_task.py clear
$ python run_task.py clear

スタックの削除 ​

これにて，今回のハンズオンは終了である． 最後にスタックを削除しよう．

スタックを削除するには，前回までと同様に， AWS コンソールにログインし CloudFormation の画面から DELETE ボタンをクリックするか，コマンドラインからコマンドを実行する． コマンドラインから行う場合は，次のコマンドを使用する．

shell
$ cdk destroy
$ cdk destroy
+ \ No newline at end of file diff --git a/development/aws/handson-serverless.html b/development/aws/handson-serverless.html index bbd4ec0c..f04f5ee1 100644 --- a/development/aws/handson-serverless.html +++ b/development/aws/handson-serverless.html @@ -5,13 +5,13 @@ Hands-on #5: サーバーレス入門 | Toshiki's Note - + - + @@ -251,9 +251,9 @@ if filename is None: filename = os.path.basename(key) - bucket.download_file(key, filename)

S3 からのダウンロードはシンプルで， download_file() メソッドを使って，ダウンロードしたい対象の Key を指定すればよい． ローカルのコンピュータでの保存先のパスを 2 個目の引数として渡している．

スタックの削除 ​

以上のハンズオンで， S3 の一番基本的な使い方を紹介した． ここまでのハンズオンが理解できたら，忘れずにスタックを削除しよう． これまでのハンズオンと同様，スタックを削除するには，次のコマンドを実行すればよい．

shell
$ cdk destroy
$ cdk destroy
- + bucket.download_file(key, filename)

S3 からのダウンロードはシンプルで， download_file() メソッドを使って，ダウンロードしたい対象の Key を指定すればよい． ローカルのコンピュータでの保存先のパスを 2 個目の引数として渡している．

スタックの削除 ​

以上のハンズオンで， S3 の一番基本的な使い方を紹介した． ここまでのハンズオンが理解できたら，忘れずにスタックを削除しよう． これまでのハンズオンと同様，スタックを削除するには，次のコマンドを実行すればよい．

shell
$ cdk destroy
$ cdk destroy
+ \ No newline at end of file diff --git a/development/aws/introduction.html b/development/aws/introduction.html index 49631ee5..39da0d1e 100644 --- a/development/aws/introduction.html +++ b/development/aws/introduction.html @@ -5,13 +5,13 @@ はじめに | Toshiki's Note - + - + @@ -33,9 +33,9 @@ -
Toshiki's Note

Appearance

TOC

はじめに ​

本書の目的・内容 ​

本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．

本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．

とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．

本書は以下のような三部構成になっている．

本書の構成
テーマハンズオン

第一部 (1章-4章)

クラウドの基礎

  • AWSに自分のサーバーを立ち上げる

第二部 (5章-9章)

クラウドを活用した機械学習

  • AWS と Jupyter を使って始めるディープラーニング

  • スケーラブルな自動質問回答ボットを作る

  • 並列化されたハイパーパラメータサーチの実装

第三部 (10章-13章)

サーバーレスアーキテクチャ入門

  • Lambda, DynamoDB, S3 の演習

  • 俳句を投稿する SNS "Bashoutter" を作る

第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．

第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．

第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．

これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．

本書のフィロソフィー ​

本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．

どういうことか？

ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．

ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．

そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．

決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．

宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)

AWS アカウント ​

本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (???) に簡単に記載したので，必要に応じて参照していただきたい．

AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．

また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．

環境構築 ​

本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (???) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．

  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (??? 参照)．

  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については ??? を参照のこと．

  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は ??? 参照のこと．

  • Node.js: version 12.0 以上 をインストールする．

  • AWS CLI: Version 2 をインストールする． インストール手順については ??? 参照のこと．

  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については ??? 参照のこと．

  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については ??? 参照のこと．

ハンズオン実行用の Docker Image ​

Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．

次のコマンドで起動する．

shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc

この Docker image の使い方や詳細は ??? に記載している．

前提知識 ​

本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．

  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．

  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．

講義に関連する資料 ​

ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．

https://github.com/tomomano/learn-aws-by-coding

本書で使用するノーテーションなど ​

  • コードやシェルのコマンドは monospace letter で記述する．

  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．

また，以下のような形式で注意やチップスを提供する．

追加のコメントなどを記す．

発展的な議論やアイディアなどを紹介する．

陥りやすいミスなどの注意事項を述べる．

絶対に犯してはならないミスを指摘する．

- +
Toshiki's Note

Appearance

TOC

はじめに ​

本書の目的・内容 ​

本書は，東京大学計数工学科で 2021 年度 S1/S2 タームに開講されている"システム情報工学特論"の講義資料として作成された．

本書の目的は，クラウドの初心者を対象とし，クラウドの基礎的な知識・概念を解説する． また， Amazon Web Services (以下， AWS) の提供するクラウド環境を実例として，具体的なクラウドの利用方法をハンズオンを通して学ぶ．

とくに，科学・エンジニアリングの学生を対象として，研究などの目的でクラウドを利用するための実践的な手順を紹介する． 知識・理論の説明は最小限に留め，実践を行う中で必要な概念の解説を行う予定である． 読者が今後，研究などでクラウドを利用する際の，足がかりとなれば本書の目的は十分達成されたことになる．

本書は以下のような三部構成になっている．

本書の構成
テーマハンズオン

第一部 (1章-4章)

クラウドの基礎

  • AWSに自分のサーバーを立ち上げる

第二部 (5章-9章)

クラウドを活用した機械学習

  • AWS と Jupyter を使って始めるディープラーニング

  • スケーラブルな自動質問回答ボットを作る

  • 並列化されたハイパーパラメータサーチの実装

第三部 (10章-13章)

サーバーレスアーキテクチャ入門

  • Lambda, DynamoDB, S3 の演習

  • 俳句を投稿する SNS "Bashoutter" を作る

第一部は，クラウドの基礎となる概念・知識を解説する． セキュリティやネットワークなど，クラウドを利用する上で最低限おさえなければいけないポイントを説明する． ハンズオンでは，はじめての仮想サーバーを AWS に立ち上げる演習を行う．

第二部では，クラウド上で科学計算 (とくに機械学習) を走らせるための入門となる知識・技術を解説する． あわせて， Docker とよばれる仮想計算環境の使用方法を紹介する． 一つ目のハンズオンでは， AWS のクラウドで Jupyter Notebook を起動し簡単な機械学習の計算を走らせる課題を実践する． 二つ目のハンズオンでは，深層学習を用いた自然言語処理により，質問に自動で回答を生成するボットを作成する． 最後に，複数台の GPU インスタンスからなるクラスターを起動し，並列に深層学習のハイパーパラメータサーチを行う方法を紹介する．

第三部では，サーバーレスアーキテクチャとよばれる最新のクラウドのアーキテクチャを紹介する． これは，サーバーの処理能力を負荷に応じてより柔軟に拡大・縮小するための概念であり，それ以前 (Serverful としばしばよばれる) と質的に異なる設計思想をクラウドに導入するものである． ハンズオンでは，サーバーレスクラウドの主要なコンポーネントである Lambda, DynamoDB, S3 の演習を提供する． さらに，サーバーレスの技術を使用して簡単な SNS をクラウド上に作成する．

これらの豊富なハンズオンにより， AWS 上にクラウドシステムを開発するための知識と技術が身につくはずである． いずれのハンズオンも，実用性を重視したものになっており，これらをベースにカスタマイズを施すことで様々な応用が可能である．

本書のフィロソフィー ​

本書のフィロソフィーを一言で表すなら， "ロケットで宇宙まで飛んでいって一度地球を眺めてみよう！" である．

どういうことか？

ここでいう"地球"とは，クラウドコンピューティングの全体像のことである． 言うまでもなく，クラウドという技術は非常に広範かつ複雑な概念で，幾多の情報技術・ハードウェア・アルゴリズムが精緻に組み合わさってできた総体である． そして，今日では科学研究から日常のインフラ設備に至るまで，我々の社会の多くの部分がクラウド技術によって支えられている．

ここでいう"ロケット"とはこの講義のことである． この講義では，ロケットに乗って宇宙まで飛び立ち，地球(クラウド)の全体を自身の目で眺めてもらう． その際，ロケットの成り立ちや仕組み (背後にある要素技術やプログラムのソースコード) を深くは問わない． 将来，自分が研究などの目的でクラウドを利用することになった際に，改めて学んでもらえば良い． 本書の目的はむしろ，クラウドの最先端に実際に触れ，そこからどんな景色が見えるか(どんな応用が可能か)を実感してもらうことである．

そのような理由で，本書はクラウドの基礎から応用まで幅広いテーマを取り扱う． 第一部はクラウドの基礎から始め，第二部では一気にレベルアップし機械学習(深層学習)をクラウドで実行する手法を解説する． さらに第三部では，サーバーレス・アーキテクチャというここ数年のうちに確立した全く新しいクラウドの設計について解説する． それぞれで本一冊分以上の内容に相当するものであるが，本書はあえてこれらを一冊にまとめ連続的に俯瞰するという野心的な意図をもって執筆された．

決して楽な搭乗体験ではないかもしれないが，このロケットにしがみついてきてもらえれば，とてもエキサイティングな景色が見られることを約束したい．

宇宙からみた地球 (Image from NASA https://www.nasa.gov/image-feature/planet-of-clouds)

AWS アカウント ​

本書では，ハンズオン形式で AWS のクラウドを実際に動かす演習を提供する． 自分でハンズオンを実行してみたい読者は，各自で AWS のアカウントの作成をしていただく． AWS のアカウントの作成の仕方は巻末付録 (???) に簡単に記載したので，必要に応じて参照していただきたい．

AWS にはいくつかの機能に対して無料利用枠が設定されており，いくつかのハンズオンは無料の範囲内で実行できる． 一方，ほかのハンズオン (とくに機械学習を扱うもの) では数ドル程度のコストが発生する． ハンズオンごとに発生するおおよそのコストについて記述があるので，注意をしながらハンズオンに取り組んでいただきたい．

また，大学などの教育機関における講義で AWS を使用する際は， AWS Educate というプログラムを利用することも可能である． これは，講義の担当者が申請を行うことで，受講する学生に対し AWS クレジットが提供されるというプログラムである． AWS Educate を利用することで金銭的な負担なしに AWS を体験することができる． また，講義を経由せず個人でも AWS Educate に参加することも可能である． AWS Educate からは様々な学習教材が提供されているので，ぜひ活用してもらいたい．

環境構築 ​

本書では， AWS 上にクラウドアプリケーションを展開するハンズオンを実施する． そこで紹介するプログラムを実行するためには，以下の計算機環境が必要である． インストールの方法については，巻末付録 (???) に記してある． 必要に応じて参照し，環境構築を各自実施していただきたい．

  • UNIX 系コンソール: ハンズオンで紹介するコマンドを実行したり， SSH でサーバーにアクセスするため， UNIX 系のコンソール環境が必要である． Mac または Linux のユーザーは， OS に標準搭載のコンソール(ターミナルとも呼ばれる)を使用すればよい． Windows のユーザーは， Windows Subsystem for Linux (WSL) を使い， Linux の仮想環境のインストールを推奨する (??? 参照)．

  • Docker: 本書では Docker とよばれる仮想計算環境の利用方法を解説する． インストール手順については ??? を参照のこと．

  • Python: Version 3.6 以上をインストールする． とくに，ハンズオンでは venv モジュールを使用する． venv の使い方は ??? 参照のこと．

  • Node.js: version 12.0 以上 をインストールする．

  • AWS CLI: Version 2 をインストールする． インストール手順については ??? 参照のこと．

  • AWS CDK: Version 1.100 以上をインストールする． Version 2 以降には未対応である． インストール手順については ??? 参照のこと．

  • AWS 認証鍵の設定: AWS API をコマンドラインから呼ぶには，認証鍵 (secret key) が設定されている必要がある． 認証鍵の設定については ??? 参照のこと．

ハンズオン実行用の Docker Image ​

Python, Node.js, AWS CDK など，ハンズオンのプログラムを実行するために必要なプログラム/ライブラリがインストール済みの Docker image を用意した． また，ハンズオンのソースコードもクローン済みである． Docker の使い方を知っている読者は，これを使えば，諸々のインストールをする必要なく，すぐにハンズオンのプログラムを実行できる．

次のコマンドで起動する．

shell
$ docker run -it tomomano/labc
$ docker run -it tomomano/labc

この Docker image の使い方や詳細は ??? に記載している．

前提知識 ​

本書を読むにあたり，要求する前提知識は大学初等程度の計算機科学の知識 (OS，プログラミングなど)のみである． それ以上の前提知識はとくに仮定しない． クラウドの利用経験もゼロで問題ない． が，以下の事前知識があるとよりスムーズに理解をすることができるだろう．

  • Python の基本的な理解: 本書では Python を使ってプログラムの作成を行う． 使用するライブラリは十分抽象化されており，関数の名前を見ただけで意味が明瞭なものがほとんどであるので， Python に詳しくなくても心配する必要はない．

  • Linux コマンドラインの基礎的な理解: クラウドを利用する際，クラウド上に立ち上がるサーバーは基本的に Linux である． Linux のコマンドラインについて知識があると，トラブルシュートなどが容易になる． 筆者のおすすめの参考書は The Linux Command Line by William Shotts である． ウェブで無料で読むことができるので，読んだことのない人はぜひ一読を．

講義に関連する資料 ​

ハンズオンで使うプログラムや教科書のソースコードは以下のウェブページで公開している．

https://github.com/tomomano/learn-aws-by-coding

本書で使用するノーテーションなど ​

  • コードやシェルのコマンドは monospace letter で記述する．

  • シェルに入力するコマンドは，それがシェルコマンドであると明示する目的で，先頭に $ がつけてある． $ はコマンドをコピー&ペーストするときは除かなければならない． 逆に，コマンドの出力には $ はついていない点に留意する．

また，以下のような形式で注意やチップスを提供する．

追加のコメントなどを記す．

発展的な議論やアイディアなどを紹介する．

陥りやすいミスなどの注意事項を述べる．

絶対に犯してはならないミスを指摘する．

+ \ No newline at end of file diff --git a/development/aws/license.html b/development/aws/license.html index 2d25fbee..fdf8cb9c 100644 --- a/development/aws/license.html +++ b/development/aws/license.html @@ -5,12 +5,12 @@ ライセンス | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

ライセンス ​

本教科書およびハンズオンのソースコードは CC BY-NC-ND 4.0 に従うライセンスで公開しています．

教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．

cc_by_nc_nd

- +
Toshiki's Note

Appearance

TOC

ライセンス ​

本教科書およびハンズオンのソースコードは CC BY-NC-ND 4.0 に従うライセンスで公開しています．

教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．

cc_by_nc_nd

+ \ No newline at end of file diff --git a/development/aws/main.html b/development/aws/main.html index d722795c..bc524734 100644 --- a/development/aws/main.html +++ b/development/aws/main.html @@ -5,7 +5,7 @@ はじめに! | Toshiki's Note - + @@ -25,7 +25,7 @@ - + @@ -1963,9 +1963,9 @@ export AWS_SECRET_ACCESS_KEY=YYYYYY export AWS_DEFAULT_REGION=ap-northeast-1

これらの環境変数は， ~/.aws/credentials よりも高い優先度をもつので，環境変数が設定されていればそちらの情報が使用される (参照)．

AWS Educate Starter Account は us-east-1 のリージョンのみ利用可能である (執筆時点での情報)． よって， AWS Educate Starter Account を使用している場合は， default region を us-east-1 に設定する必要がある．

AWS CDK のインストール ​

読者のために，執筆時点におけるインストールの手順 (Linux 向け) を簡単に記述する． 将来のバージョンでは変更される可能性があるので，常に 公式のドキュメンテーション で最新の情報をチェックすることを忘れずに．

Node.js がインストールされていれば，基本的に次のコマンドを実行すればよい．

shell
$ sudo npm install -g aws-cdk
$ sudo npm install -g aws-cdk

本書のハンズオンは AWS CDK version 1.100.0 で開発した． CDK は開発途上のライブラリなので，将来的に API が変更される可能性がある． API の変更によりエラーが生じた場合は， version 1.100.0 を使用することを推奨する．

shell
$ npm install -g aws-cdk@1.100
$ npm install -g aws-cdk@1.100

インストールできたか確認するため，次のコマンドを打って正しくバージョンが表示されることを確認する．

shell
$ cdk --version
$ cdk --version

インストールができたら，次のコマンドにより AWS 側の初期設定を行う． これは一度実行すれば OK．

shell
$ cdk bootstrap
$ cdk bootstrap

cdk bootstrap を実行するときは，AWS の認証情報とリージョンが正しく設定されていることを確認する． デフォルトでは ~/.aws/config にあるデフォルトのプロファイルが使用される． デフォルト以外のプロファイルを用いるときは AWS CLI のインストール で紹介したテクニックを使って切り替える．

AWS CDK の認証情報の設定は AWS CLI と基本的に同じである．詳しくは AWS CLI のインストール を参照．

WSL のインストール ​

本書のハンズオンではコマンドラインから AWS CLI のコマンドを実行したり， Python で書かれたプログラムを実行する． コマンドは基本的に UNIX のターミナルを想定して書かれている． Linux や Mac のユーザーは OS に標準搭載されているターミナルを用いれば良い． Windows を利用している読者は， Windows Subsystem for Linux (WSL) を利用することで，仮想の Linux 環境を構築することを推奨する． Cygwin などの Linux 環境をエミュレートするほかのツールでも構わないが，本書のプログラムは WSL でのみ動作確認を行っている．

WSL とは， Windows の OS 上で Linux の仮想環境を起動するための， Microsoft 社が公式で提供しているソフトウェアである． Ubuntu など希望の Linux distribution が選択でき，基本的にすべての Linux 向けに作られたプログラム・ソフトウェアを使用することができる．

執筆時点では WSL 2 が最新版として提供されているので，以下では WSL 2 のインストール手順を簡単に説明する． 細かな詳細などは， 公式ドキュメンテーション を参照のこと．

前提として，使用される OS は Windows 10 (Pro または Home エディション) でなければならない． さらに，使用している Windows 10 のバージョンが WSL に対応するバージョンであるかを確認する． X64 のシステムでは Version 1903, Build 18362 以上でなければならない． バージョンが対応していない場合は、 Windows のアップデートを行う．

まず最初に， Administrator 権限で PowerShell を起動する (figure_title)． 左下の Windows メニューの検索バーに powershell と入力すると， PowerShell のプログラムが見つかるはずである， これを右クリックし、 Run as administrator を選択し起動する．

管理者権限での PowerShell の起動

PowerShell が起動したら、次のコマンドを実行する．

powershell
dism.exe /online /enable-feature /featurename:Microsoft-Windows-Subsystem-Linux /all /norestart
dism.exe /online /enable-feature /featurename:Microsoft-Windows-Subsystem-Linux /all /norestart

実行して、“The operation completed successfully.” と出力されるのを確認する． これで WSL が enable される．

次に，先ほどと同じ Administrator 権限で開いた PowerShell で次のコマンドを実行する。

powersh
dism.exe /online /enable-feature /featurename:VirtualMachinePlatform /all /norestart
dism.exe /online /enable-feature /featurename:VirtualMachinePlatform /all /norestart

実行して， “The operation completed successfully.” と出力されるのを確認する． これが確認出来たら、一度コンピュータを再起動する．

続いて， Linux kernel update package を次のリンクからダウンロードする． https://wslstorestorage.blob.core.windows.net/wslblob/wsl_update_x64.msi

ダウンロードしたファイルをダブルクリックして実行する． ダイアログに従ってインストールを完了させる．

そうしたら，再び PowerShell を開き次のコマンドを実行する。

shell
wsl --set-default-version 2
wsl --set-default-version 2

最後に、自分の好みの Linux distribution をインストールする． ここでは Ubuntu 20.04 をインストールしよう．

Microsoft store のアプリを起動し，検索バーに Ubuntu と入力する． Ubuntu 20.04 LTS という項目が見つかるはずなので，それを開き， “Get” ボタンをクリックする (figure_title)． しばらく待つと， Ubuntu 20.04 のインストールが完了する．

Microsoft store から Ubuntu 20.04 をインストール

Ubuntu 20.04 を初回に起動すると，初期設定が自動で開始され，数分待つことになる． 初期設定が終わると，ユーザー名・パスワードを設定するようプロンプトが出るので，プロンプトに従い入力する．

これで WSL2 のインストールが完了した． 早速 WSL2 を起動してみよう． 左下の Windows メニューの検索バーに Ubuntu と入力すると， Ubuntu 20.04 のプログラムが見つかるはずである (figure_title)． クリックして起動しよう．

Ubuntu 20.04 の起動

すると，ターミナルの黒い画面が立ち上がるだろう (figure_title)． ls, top などのコマンドを打ってみて， WSL がきちんと動作していることを確認しよう．

WSL の起動画面

オプションとして， Windows Terminal というマイクロソフトから提供されているツールを使うと，より快適に WSL を使用することができる． 興味のある読者はこちらのインストールも推奨する．

Docker のインストール ​

Docker のインストールの方法は OS によって異なる．

Mac ユーザーは， Docker Desktop をインストールする． インストールの方法は， Docker のウェブサイト から， Mac 版の Docker Desktop をダウンロードし，ダウンロードされたファイルをダブルクリックし， Applications のフォルダにドラッグするだけで良い． 詳細は 公式ドキュメンテーション を参照のこと．

Windows のユーザーは，Docker Desktop をインストールする． その際， WSL 2 が事前にインストールされていなければならない． 詳細は 公式ドキュメンテーション を参照のこと． Docker Desktop をインストールすると， WSL からも docker コマンドが使用できるようになる．

Linux ユーザー (特に Ubuntu ユーザー) については，インストールの方法はいくつかのアプローチがある． 公式ドキュメンテーション にいくつかのインストールの方法が示されているので，詳しい情報はそちらを参照いただきたい．

最も簡単な方法は， Docker が公式で提供しているインストールスクリプトを用いる方法である． この場合，次のコマンドを実行することで Docker がインストールされる．

shell
$ curl -fsSL https://get.docker.com -o get-docker.sh
 $ sudo sh get-docker.sh
$ curl -fsSL https://get.docker.com -o get-docker.sh
-$ sudo sh get-docker.sh

デフォルトのインストールでは， root ユーザーのみが docker コマンドを使用できる設定になっている． 従って，コマンドには毎回 sudo を付け加える必要がある． これが面倒だと感じる場合は，次のステップにより，使用するユーザーを docker というグループに追加する (詳細は 公式ドキュメンテーション "Post-installation steps for Linux" を参照)．

まず最初に， docker という名前にグループを追加する． インストールによっては，既に docker グループが作られている場合もある．

sh
$ sudo groupadd docker
$ sudo groupadd docker

次に，現在使用しているユーザーを docker グループに加える．

sh
$ sudo usermod -aG docker $USER
$ sudo usermod -aG docker $USER

ここまでできたら，一度ログアウトし，再度ログインする． これによって，グループの変更がターミナルのセッションに反映される．

設定が正しくできているかを確認するため，次のコマンドを実行してみる．

sh
$ docker run hello-world
$ docker run hello-world

sudo なしでコンテナが実行できたならば，設定は完了である．

Python venv クイックガイド ​

他人からもらったプログラムで， numpy や scipy のバージョンが違う！などの理由で，プログラムが動かない，という経験をしたことがある人は多いのではないだろうか． もし，自分の計算機の中に一つしか Python 環境がないとすると，プロジェクトを切り替えるごとに正しいバージョンをインストールし直さなければならず，これは大変な手間である．

コードのシェアをよりスムーズにするためには，ライブラリのバージョンはプロジェクトごとに管理されるべきである． それを可能にするのが Python 仮想環境とよばれるツールであり， venv, pyenv, conda などがよく使われる．

そのなかでも， venv は Python に標準搭載されているのでとても便利である． pyenv や conda は，別途インストールの必要があるが，それぞれの長所もある．

venv を使って仮想環境を作成するには，

sh
$ python -m venv .env
$ python -m venv .env

と実行する． これにより .env/ というディレクトリが作られ，このディレクトリに依存するライブラリが保存されることになる．

この新たな仮想環境を起動するには

sh
$ source .env/bin/activate
$ source .env/bin/activate

と実行する．

シェルのプロンプトに (.env) という文字が追加されていることを確認しよう (figure_title)． これが， "いまあなたは venv の中にいますよ" というしるしになる．

venv を起動したときのプロンプト

仮想環境を起動すると，それ以降実行する pip コマンドは， .env/ 以下にインストールされる．このようにして，プロジェクトごとに使うライブラリのバージョンを切り分けることができる．

Python では requirements.txt というファイルに依存ライブラリを記述するのが一般的な慣例である．他人からもらったプログラムに， requirements.txt が定義されていれば，

sh
$ pip install -r requirements.txt
$ pip install -r requirements.txt

と実行することで，必要なライブラリをインストールし，瞬時に Python 環境を再現することができる．

venv による仮想環境を保存するディレクトリの名前は任意に選べることができるが， .env という名前を用いるのが一般的である．

ハンズオン実行用の Docker image の使い方 ​

ハンズオンを実行するために必要な， Node.js, Python, AWS CDK などがインストールされた Docker image を用意した． これを使用することで，自分のローカルマシンに諸々をインストールする必要なく，すぐにハンズオンのコードが実行できる．

ハンズオンのいくつかのコマンドは Docker の外 = ローカルマシンのリアル環境で実行されなければならない． それらについてはハンズオンの該当箇所に注意書きとして記してある．

Docker イメージは Docker Hub においてある． Docker イメージのビルドファイルは GitHub の docker/Dockerfile にある．

次のコマンドでコンテナを起動する．

sh
$ docker run -it tomomano/labc:latest
$ docker run -it tomomano/labc:latest

初回にコマンドを実行したときのみ，イメージが Docker Hub からダウンロード (pull) される． 二回目以降はローカルにダウンロードされたイメージが使用される．

コンテナが起動すると，次のようなインタラクティブシェルが表示されるはずである (起動時に -it のオプションをつけたのがポイントである)．

sh
root@aws-handson:~/programlisting
root@aws-handson:~/programlisting

この状態で ls コマンドを打つと， handson/ というディレクトリがあるはずである． ここに cd する．

sh
$ cd handson
$ cd handson

すると，各ハンズオンごとのディレクトリが見つかるはずである．

あとは，ハンズオンごとにディレクトリを移動し，ハンズオンごとの virtualenv を作成し，スタックのデプロイを行えばよい (プログラムを実行する など参照)． ハンズオンごとに使用する依存ライブラリが異なるので，それぞれのハンズオンごとに virtualenv を作成するという設計になっている．

AWS の認証情報を設定することも忘れずに． AWS CLI のインストール で記述したように， AWS_ACCESS_KEY_ID などの環境変数を設定するのが簡単な方法である． あるいは，ローカルマシンの ~/.aws/credentials に認証情報が書き込まれているなら，このディレクトリをコンテナにマウントすることで，同じ認証ファイルをコンテナ内部から参照することが可能である． この選択肢を取る場合は，次のコマンドでコンテナを起動する．

sh
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest

これにより，ローカルマシンの ~/.aws をコンテナの /root/.aws にマウントすることができる． 最後の :ro は read-only を意味する． 大切な認証ファイルが誤って書き換えられてしまわないように， read-only のフラグをつけることをおすすめする．

/root/ がコンテナ環境におけるホームディレクトリである． ここで紹介した認証ファイルをマウントするテクニックは， SSH 鍵をコンテナに渡すときなどにも使える．

ライセンス ​

本教科書およびハンズオンのソースコードは CC BY-NC-ND 4.0 に従うライセンスで公開しています．

教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．

- +$ sudo sh get-docker.sh

デフォルトのインストールでは， root ユーザーのみが docker コマンドを使用できる設定になっている． 従って，コマンドには毎回 sudo を付け加える必要がある． これが面倒だと感じる場合は，次のステップにより，使用するユーザーを docker というグループに追加する (詳細は 公式ドキュメンテーション "Post-installation steps for Linux" を参照)．

まず最初に， docker という名前にグループを追加する． インストールによっては，既に docker グループが作られている場合もある．

sh
$ sudo groupadd docker
$ sudo groupadd docker

次に，現在使用しているユーザーを docker グループに加える．

sh
$ sudo usermod -aG docker $USER
$ sudo usermod -aG docker $USER

ここまでできたら，一度ログアウトし，再度ログインする． これによって，グループの変更がターミナルのセッションに反映される．

設定が正しくできているかを確認するため，次のコマンドを実行してみる．

sh
$ docker run hello-world
$ docker run hello-world

sudo なしでコンテナが実行できたならば，設定は完了である．

Python venv クイックガイド ​

他人からもらったプログラムで， numpy や scipy のバージョンが違う！などの理由で，プログラムが動かない，という経験をしたことがある人は多いのではないだろうか． もし，自分の計算機の中に一つしか Python 環境がないとすると，プロジェクトを切り替えるごとに正しいバージョンをインストールし直さなければならず，これは大変な手間である．

コードのシェアをよりスムーズにするためには，ライブラリのバージョンはプロジェクトごとに管理されるべきである． それを可能にするのが Python 仮想環境とよばれるツールであり， venv, pyenv, conda などがよく使われる．

そのなかでも， venv は Python に標準搭載されているのでとても便利である． pyenv や conda は，別途インストールの必要があるが，それぞれの長所もある．

venv を使って仮想環境を作成するには，

sh
$ python -m venv .env
$ python -m venv .env

と実行する． これにより .env/ というディレクトリが作られ，このディレクトリに依存するライブラリが保存されることになる．

この新たな仮想環境を起動するには

sh
$ source .env/bin/activate
$ source .env/bin/activate

と実行する．

シェルのプロンプトに (.env) という文字が追加されていることを確認しよう (figure_title)． これが， "いまあなたは venv の中にいますよ" というしるしになる．

venv を起動したときのプロンプト

仮想環境を起動すると，それ以降実行する pip コマンドは， .env/ 以下にインストールされる．このようにして，プロジェクトごとに使うライブラリのバージョンを切り分けることができる．

Python では requirements.txt というファイルに依存ライブラリを記述するのが一般的な慣例である．他人からもらったプログラムに， requirements.txt が定義されていれば，

sh
$ pip install -r requirements.txt
$ pip install -r requirements.txt

と実行することで，必要なライブラリをインストールし，瞬時に Python 環境を再現することができる．

venv による仮想環境を保存するディレクトリの名前は任意に選べることができるが， .env という名前を用いるのが一般的である．

ハンズオン実行用の Docker image の使い方 ​

ハンズオンを実行するために必要な， Node.js, Python, AWS CDK などがインストールされた Docker image を用意した． これを使用することで，自分のローカルマシンに諸々をインストールする必要なく，すぐにハンズオンのコードが実行できる．

ハンズオンのいくつかのコマンドは Docker の外 = ローカルマシンのリアル環境で実行されなければならない． それらについてはハンズオンの該当箇所に注意書きとして記してある．

Docker イメージは Docker Hub においてある． Docker イメージのビルドファイルは GitHub の docker/Dockerfile にある．

次のコマンドでコンテナを起動する．

sh
$ docker run -it tomomano/labc:latest
$ docker run -it tomomano/labc:latest

初回にコマンドを実行したときのみ，イメージが Docker Hub からダウンロード (pull) される． 二回目以降はローカルにダウンロードされたイメージが使用される．

コンテナが起動すると，次のようなインタラクティブシェルが表示されるはずである (起動時に -it のオプションをつけたのがポイントである)．

sh
root@aws-handson:~/programlisting
root@aws-handson:~/programlisting

この状態で ls コマンドを打つと， handson/ というディレクトリがあるはずである． ここに cd する．

sh
$ cd handson
$ cd handson

すると，各ハンズオンごとのディレクトリが見つかるはずである．

あとは，ハンズオンごとにディレクトリを移動し，ハンズオンごとの virtualenv を作成し，スタックのデプロイを行えばよい (プログラムを実行する など参照)． ハンズオンごとに使用する依存ライブラリが異なるので，それぞれのハンズオンごとに virtualenv を作成するという設計になっている．

AWS の認証情報を設定することも忘れずに． AWS CLI のインストール で記述したように， AWS_ACCESS_KEY_ID などの環境変数を設定するのが簡単な方法である． あるいは，ローカルマシンの ~/.aws/credentials に認証情報が書き込まれているなら，このディレクトリをコンテナにマウントすることで，同じ認証ファイルをコンテナ内部から参照することが可能である． この選択肢を取る場合は，次のコマンドでコンテナを起動する．

sh
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest
$ docker run -it -v ~/.aws:/root/.aws:ro tomomano/labc:latest

これにより，ローカルマシンの ~/.aws をコンテナの /root/.aws にマウントすることができる． 最後の :ro は read-only を意味する． 大切な認証ファイルが誤って書き換えられてしまわないように， read-only のフラグをつけることをおすすめする．

/root/ がコンテナ環境におけるホームディレクトリである． ここで紹介した認証ファイルをマウントするテクニックは， SSH 鍵をコンテナに渡すときなどにも使える．

ライセンス ​

本教科書およびハンズオンのソースコードは CC BY-NC-ND 4.0 に従うライセンスで公開しています．

教育など非商用の目的での本教科書の使用や再配布は自由に行うことが可能です． 商用目的で本書の全体またはその一部を無断で転載する行為は，これを固く禁じます．

+ \ No newline at end of file diff --git a/development/aws/scientific-computing.html b/development/aws/scientific-computing.html index 5db27105..035afb20 100644 --- a/development/aws/scientific-computing.html +++ b/development/aws/scientific-computing.html @@ -5,13 +5,13 @@ クラウドで行う科学計算・機械学習 | Toshiki's Note - + - + @@ -33,9 +33,9 @@ -
Toshiki's Note

Appearance

TOC

クラウドで行う科学計算・機械学習 ​

計算機が発達した現代では，計算機によるシミュレーションやビッグデータの解析は，科学・エンジニアリングの研究の主要な柱である． これらの大規模な計算を実行するには，クラウドは最適である． 本章から始まる第二部では，どのようにしてクラウド上で科学計算を実行するのかを，ハンズオンとともに体験してもらう． 科学計算の具体的な題材として，今回は機械学習(深層学習)を取り上げる．

なお，本書では PyTorch ライブラリを使って深層学習のアルゴリズムを実装するが，深層学習および PyTorch の知識は不要である． 講義ではなぜ・どうやって深層学習をクラウドで実行するかに主眼を置いているので，実行するプログラムの詳細には立ち入らない． 将来，自分で深層学習を使う機会が来たときに，詳しく学んでもらいたい．

なぜ機械学習をクラウドで行うのか？ ​

2010 年頃に始まった第三次 AI ブームのおかげで，学術研究だけでなく社会・ビジネスの文脈でも機械学習に高い関心が寄せられている． とくに，深層学習 (ディープラーニング) とよばれる多層のレイヤーからなるニューラルネットワークを用いたアルゴリズムは，画像認識や自然言語処理などの分野で圧倒的に高い性能を実現し，革命をもたらしている．

深層学習の特徴は，なんといってもそのパラメータの多さである． 層が深くなるほど，層間のニューロンを結ぶ重みパラメータの数が増大していく． たとえば，最新の言語モデルである GPT-3 には1750 億個ものパラメータが含まれている． このような膨大なパラメータを有することで，深層学習は高い表現力と汎化性能を実現しているのである．

GPT-3 に限らず，最近の SOTA (State-of-the-art) の性能を達成するニューラルネットワークでは，百万から億のオーダーのパラメータを内包することは頻繁になってきている． そのような巨大なニューラルネットを訓練 (最適化) させるのは，当然のことながら膨大な計算コストがかかる． 結果として，ひとつの計算機では丸一日以上の時間がかかる場合も珍しくない． 深層学習の発展の速度は目覚ましく，研究・ビジネス両方の観点からも，いかにスループットよくニューラルネットワークの最適化を行えるかが鍵となってくる． そのような問題を解決するのにとても有効な手段が，クラウドである！ ??? でその片鱗を見たように，クラウドを使用することでゼロから数千に至るまでの数のインスタンスを動的に起動し，並列に計算を実行することができる． さらに，深層学習を加速させる目的で，深層学習の演算に専用設計された計算チップ (GPU など) がある． クラウドを利用すると，そのような専用計算チップも無尽蔵に利用することができる． 事実，先述した GPT-3 の学習も，詳細は明かされていないが，Microsoft 社のクラウドを使って行われたと報告されている．

GPU による深層学習の高速化 ​

深層学習の計算で欠かすことのできない技術として， GPU (Graphics Processing Unit) について少し説明する．

GPU は，その名のとおり，元々はコンピュータグラフィックスを出力するための専用計算チップである． CPU (Central Processing Unit) に対し，グラフィックスの演算に特化した設計がなされている． 身近なところでは， XBox や PS5 などのゲームコンソールなどに搭載されているし，ハイエンドなノート型・デスクトップ型計算機にも搭載されていることがある． コンピュータグラフィックスでは，スクリーンにアレイ状に並んだ数百万個の画素をビデオレート (30 fps) 以上で処理する必要がある． そのため，GPU はコアあたりの演算能力は比較的小さいかわりに，チップあたり数百から数千のコアを搭載しており (figure_title)，スクリーンの画素を並列的に処理することで，リアルタイムでの描画を実現している．

GPUのアーキテクチャ．GPUには数百から数千の独立した計算コアが搭載されている． (画像出典: https://devblogs.nvidia.com/nvidia-turing-architecture-in-depth/)

このように，コンピュータグラフィクスの目的で生まれた GPU だが，2010 年前後から，その高い並列計算能力をグラフィックス以外の計算 (科学計算など) に用いるという流れ (General-purpose computing on GPU; GPGPU) が生まれた． GPU のコアは，その設計から，行列の計算など，単純かつ規則的な演算が得意であり，そのような演算に対しては数個程度のコアしかもたない CPU に比べて圧倒的に高い計算速度を実現することができる． 現在では GPGPU は分子動力学や気象シミュレーション，そして機械学習など多くの分野で使われている．

ディープラーニングで最も頻繁に起こる演算が，ニューロンの出力を次の層のニューロンに伝える畳み込み (Convolution) 演算である (figure_title)． 畳み込み演算は，まさに GPU が得意とする演算であり， CPU ではなく GPU を用いることで学習を飛躍的に (最大で数百倍程度) 加速させることができる．

ニューラルネットワークにおける畳み込み演算．

このように GPU は機械学習の計算で欠かせないものであるが，なかなか高価である． たとえば，科学計算・機械学習に専用設計された NVIDIA 社の Tesla V100 というチップは，一台で約百万円の価格が設定されている． 機械学習を始めるのに，いきなり百万円の投資はなかなか大きい． だが，クラウドを使えば，初期コスト０で GPU を使用することができる．

機械学習を行うのに， V100 が必ずしも必要というわけではない． むしろ，研究者などでしばしば行われるのは，コンピュータゲームに使われるグラフィックス用の GPU を買ってきて (NVIDIA GeForce シリーズなど)，開発のときはをそれを用いる，というアプローチである． グラフィックス用のいわゆる"コンシューマ GPU"は，市場の需要が大きいおかげで，10 万円前後の価格で購入することができる． V100 と比べると，コンシューマ GPU はコアの数が少なかったり，メモリーが小さかったりなどで劣る点があるが， それらを除いては計算能力にとくに制限があるわけではなく，開発の段階では十分な性能である場合がほとんどである． プログラムができあがって，ビッグデータの解析や，モデルをさらに大きくしたいときなどに，クラウドは有効だろう．

クラウドで GPU を使うには， GPU が搭載された EC2 インスタンスタイプ (P3, P2, G3, G4 など) を選択しなければならない． table_title に，代表的な GPU 搭載のインスタンスタイプを挙げる (執筆時点での情報)．

GPUを搭載したEC2インスタンスタイプ
InstanceGPUsGPU modelGPU Mem (GiB)vCPUMem (GiB)Price per hour ($)

p3.2xlarge

1

NVIDIA V100

16

8

61

3.06

p3n.16xlarge

8

NVIDIA V100

128

64

488

24.48

p2.xlarge

1

NVIDIA K80

12

4

61

0.9

g4dn.xlarge

1

NVIDIA T4

16

4

16

0.526

table_title からわかるとおり， CPU のみのインスタンスと比べると少し高い価格設定になっている． また，古い世代の GPU (V100 に対しての K80) はより安価な価格で提供されている． 1 インスタンスあたりの GPU の搭載数は 1 台から最大で 8 台まで選択することが可能である．

GPU を搭載した一番安いインスタンスタイプは， g4dn.xlarge であり，これには廉価かつ省エネルギー設計の NVIDIA T4 が搭載されている． 後のハンズオンでは，このインスタンスを使用して，ディープラーニングの計算を行ってみる．

table_title の価格は us-east-1 のものである． リージョンによって多少価格設定が異なる．

V100 を一台搭載した p3.2xlarge の利用料金は一時間あたり $3.06 である． V100 が約百万円で売られていることを考えると，約 3000 時間 (= 124 日間)，通算で計算を行った場合に，クラウドを使うよりも V100 を自分で買ったほうがお得になる，という計算になる (実際には，自前で V100 を用意する場合は， V100 だけでなく， CPU やネットワーク機器，電気使用料も必要なので，百万円よりもさらにコストがかかる)．

GPT-3 で使われた計算リソースの詳細は論文でも明かされていないのだが， Lambda 社のブログで興味深い考察が行われている (Lambda 社は機械学習に特化したクラウドサービスを提供している)．

記事によると，1750 億のパラメータを訓練するには，一台の GPU (NVIDIA V100) を用いた場合，342 年の月日と 460 万ドルのクラウド利用料が必要となる，とのことである． GPT-3 のチームは，複数の GPU に処理を分散することで現実的な時間のうちに訓練を完了させたのであろうが，このレベルのモデルになってくるとクラウド技術の限界を攻めないと達成できないことは確かである．

深層学習を詳しく勉強したい人には以下の参考書を推薦したい． 深層学習の基礎的な概念や理論は普遍的であるが，この分野は日進月歩なので，常に最新の情報を取り入れることを忘れずに．

- +
Toshiki's Note

Appearance

TOC

クラウドで行う科学計算・機械学習 ​

計算機が発達した現代では，計算機によるシミュレーションやビッグデータの解析は，科学・エンジニアリングの研究の主要な柱である． これらの大規模な計算を実行するには，クラウドは最適である． 本章から始まる第二部では，どのようにしてクラウド上で科学計算を実行するのかを，ハンズオンとともに体験してもらう． 科学計算の具体的な題材として，今回は機械学習(深層学習)を取り上げる．

なお，本書では PyTorch ライブラリを使って深層学習のアルゴリズムを実装するが，深層学習および PyTorch の知識は不要である． 講義ではなぜ・どうやって深層学習をクラウドで実行するかに主眼を置いているので，実行するプログラムの詳細には立ち入らない． 将来，自分で深層学習を使う機会が来たときに，詳しく学んでもらいたい．

なぜ機械学習をクラウドで行うのか？ ​

2010 年頃に始まった第三次 AI ブームのおかげで，学術研究だけでなく社会・ビジネスの文脈でも機械学習に高い関心が寄せられている． とくに，深層学習 (ディープラーニング) とよばれる多層のレイヤーからなるニューラルネットワークを用いたアルゴリズムは，画像認識や自然言語処理などの分野で圧倒的に高い性能を実現し，革命をもたらしている．

深層学習の特徴は，なんといってもそのパラメータの多さである． 層が深くなるほど，層間のニューロンを結ぶ重みパラメータの数が増大していく． たとえば，最新の言語モデルである GPT-3 には1750 億個ものパラメータが含まれている． このような膨大なパラメータを有することで，深層学習は高い表現力と汎化性能を実現しているのである．

GPT-3 に限らず，最近の SOTA (State-of-the-art) の性能を達成するニューラルネットワークでは，百万から億のオーダーのパラメータを内包することは頻繁になってきている． そのような巨大なニューラルネットを訓練 (最適化) させるのは，当然のことながら膨大な計算コストがかかる． 結果として，ひとつの計算機では丸一日以上の時間がかかる場合も珍しくない． 深層学習の発展の速度は目覚ましく，研究・ビジネス両方の観点からも，いかにスループットよくニューラルネットワークの最適化を行えるかが鍵となってくる． そのような問題を解決するのにとても有効な手段が，クラウドである！ ??? でその片鱗を見たように，クラウドを使用することでゼロから数千に至るまでの数のインスタンスを動的に起動し，並列に計算を実行することができる． さらに，深層学習を加速させる目的で，深層学習の演算に専用設計された計算チップ (GPU など) がある． クラウドを利用すると，そのような専用計算チップも無尽蔵に利用することができる． 事実，先述した GPT-3 の学習も，詳細は明かされていないが，Microsoft 社のクラウドを使って行われたと報告されている．

GPU による深層学習の高速化 ​

深層学習の計算で欠かすことのできない技術として， GPU (Graphics Processing Unit) について少し説明する．

GPU は，その名のとおり，元々はコンピュータグラフィックスを出力するための専用計算チップである． CPU (Central Processing Unit) に対し，グラフィックスの演算に特化した設計がなされている． 身近なところでは， XBox や PS5 などのゲームコンソールなどに搭載されているし，ハイエンドなノート型・デスクトップ型計算機にも搭載されていることがある． コンピュータグラフィックスでは，スクリーンにアレイ状に並んだ数百万個の画素をビデオレート (30 fps) 以上で処理する必要がある． そのため，GPU はコアあたりの演算能力は比較的小さいかわりに，チップあたり数百から数千のコアを搭載しており (figure_title)，スクリーンの画素を並列的に処理することで，リアルタイムでの描画を実現している．

GPUのアーキテクチャ．GPUには数百から数千の独立した計算コアが搭載されている． (画像出典: https://devblogs.nvidia.com/nvidia-turing-architecture-in-depth/)

このように，コンピュータグラフィクスの目的で生まれた GPU だが，2010 年前後から，その高い並列計算能力をグラフィックス以外の計算 (科学計算など) に用いるという流れ (General-purpose computing on GPU; GPGPU) が生まれた． GPU のコアは，その設計から，行列の計算など，単純かつ規則的な演算が得意であり，そのような演算に対しては数個程度のコアしかもたない CPU に比べて圧倒的に高い計算速度を実現することができる． 現在では GPGPU は分子動力学や気象シミュレーション，そして機械学習など多くの分野で使われている．

ディープラーニングで最も頻繁に起こる演算が，ニューロンの出力を次の層のニューロンに伝える畳み込み (Convolution) 演算である (figure_title)． 畳み込み演算は，まさに GPU が得意とする演算であり， CPU ではなく GPU を用いることで学習を飛躍的に (最大で数百倍程度) 加速させることができる．

ニューラルネットワークにおける畳み込み演算．

このように GPU は機械学習の計算で欠かせないものであるが，なかなか高価である． たとえば，科学計算・機械学習に専用設計された NVIDIA 社の Tesla V100 というチップは，一台で約百万円の価格が設定されている． 機械学習を始めるのに，いきなり百万円の投資はなかなか大きい． だが，クラウドを使えば，初期コスト０で GPU を使用することができる．

機械学習を行うのに， V100 が必ずしも必要というわけではない． むしろ，研究者などでしばしば行われるのは，コンピュータゲームに使われるグラフィックス用の GPU を買ってきて (NVIDIA GeForce シリーズなど)，開発のときはをそれを用いる，というアプローチである． グラフィックス用のいわゆる"コンシューマ GPU"は，市場の需要が大きいおかげで，10 万円前後の価格で購入することができる． V100 と比べると，コンシューマ GPU はコアの数が少なかったり，メモリーが小さかったりなどで劣る点があるが， それらを除いては計算能力にとくに制限があるわけではなく，開発の段階では十分な性能である場合がほとんどである． プログラムができあがって，ビッグデータの解析や，モデルをさらに大きくしたいときなどに，クラウドは有効だろう．

クラウドで GPU を使うには， GPU が搭載された EC2 インスタンスタイプ (P3, P2, G3, G4 など) を選択しなければならない． table_title に，代表的な GPU 搭載のインスタンスタイプを挙げる (執筆時点での情報)．

GPUを搭載したEC2インスタンスタイプ
InstanceGPUsGPU modelGPU Mem (GiB)vCPUMem (GiB)Price per hour ($)

p3.2xlarge

1

NVIDIA V100

16

8

61

3.06

p3n.16xlarge

8

NVIDIA V100

128

64

488

24.48

p2.xlarge

1

NVIDIA K80

12

4

61

0.9

g4dn.xlarge

1

NVIDIA T4

16

4

16

0.526

table_title からわかるとおり， CPU のみのインスタンスと比べると少し高い価格設定になっている． また，古い世代の GPU (V100 に対しての K80) はより安価な価格で提供されている． 1 インスタンスあたりの GPU の搭載数は 1 台から最大で 8 台まで選択することが可能である．

GPU を搭載した一番安いインスタンスタイプは， g4dn.xlarge であり，これには廉価かつ省エネルギー設計の NVIDIA T4 が搭載されている． 後のハンズオンでは，このインスタンスを使用して，ディープラーニングの計算を行ってみる．

table_title の価格は us-east-1 のものである． リージョンによって多少価格設定が異なる．

V100 を一台搭載した p3.2xlarge の利用料金は一時間あたり $3.06 である． V100 が約百万円で売られていることを考えると，約 3000 時間 (= 124 日間)，通算で計算を行った場合に，クラウドを使うよりも V100 を自分で買ったほうがお得になる，という計算になる (実際には，自前で V100 を用意する場合は， V100 だけでなく， CPU やネットワーク機器，電気使用料も必要なので，百万円よりもさらにコストがかかる)．

GPT-3 で使われた計算リソースの詳細は論文でも明かされていないのだが， Lambda 社のブログで興味深い考察が行われている (Lambda 社は機械学習に特化したクラウドサービスを提供している)．

記事によると，1750 億のパラメータを訓練するには，一台の GPU (NVIDIA V100) を用いた場合，342 年の月日と 460 万ドルのクラウド利用料が必要となる，とのことである． GPT-3 のチームは，複数の GPU に処理を分散することで現実的な時間のうちに訓練を完了させたのであろうが，このレベルのモデルになってくるとクラウド技術の限界を攻めないと達成できないことは確かである．

深層学習を詳しく勉強したい人には以下の参考書を推薦したい． 深層学習の基礎的な概念や理論は普遍的であるが，この分野は日進月歩なので，常に最新の情報を取り入れることを忘れずに．

+ \ No newline at end of file diff --git a/development/aws/serverless.html b/development/aws/serverless.html index 7a7d1153..9c913245 100644 --- a/development/aws/serverless.html +++ b/development/aws/serverless.html @@ -5,13 +5,13 @@ Serverless architecture | Toshiki's Note - + - + @@ -33,9 +33,9 @@ -
Toshiki's Note

Appearance

TOC

Serverless architecture ​

サーバーレスアーキテクチャ (Serverless architecture) あるいは サーバーレスコンピューティング (Serverless computing) とは，従来とは全く異なるアプローチに基づくクラウドシステムの設計方法である． 歴史的には， AWS が 2014 年に発表した Lamba がサーバーレスアーキテクチャの先駆けとされている． その後， Google や Microsoft などのクラウドプラットフォームも同様の機能の提供を開始している． サーバーレスアーキテクチャの利点は，スケーラブルなクラウドシステムを安価かつ簡易に作成できる点であり，近年いたるところで導入が進んでいる．

Serverless とは，文字どおりの意味としてはサーバーなしで計算をするということになるが，それは一体どういう意味だろうか？ サーバーレスについて説明するためには，まずは従来的な， "serverful" とよばれるようなシステムについて解説しなければならない．

Serverful クラウド (従来型) ​

従来的なクラウドシステムのスケッチを figure_title に示す． クライアントから送信されたリクエストは，最初に API サーバーに送られる． API サーバーでは，リクエストの内容に応じてタスクが実行される． タスクには，API サーバーだけで完結できるものもあるが，多くの場合，データベースの読み書きが必要である． データベースには，データベース専用の独立したサーバーマシンが用いられることが一般的である． また，画像や動画などの容量の大きいデータは，また別のストレージサーバーに保存されることが多い． これらの API サーバー，データベースサーバー，ストレージサーバーはそれぞれ独立したサーバーマシンであり， AWS の言葉では EC2 による仮想インスタンスを想定してもらったらよい．

多くのウェブサービスでは，多数のクライアントからのリクエストを処理するため，複数のサーバーマシンがクラウド内で起動し，負荷を分散するような設計がなされている． クライアントから来たリクエストを計算容量に余裕のあるサーバーに振り分けるような操作を Load balancing とよび，そのような操作を担当するマシンのことを Load balancer という．

計算負荷を分散する目的で多数のインスタンスを起動するのはよいのだが，計算負荷が小さすぎてアイドリング状態にあるようではコストと電力の無駄遣いである． したがって，すべてのサーバーが常に目標とする計算負荷を維持するよう，計算の負荷に応じてクラスター内の仮想サーバーの数を動的に増減させるような仕組みが必要である． そのような仕組みをクラスターのスケーリングとよび，負荷の増大に応答して新しい仮想インスタンスをクラスターに追加する操作を scale-out，負荷の減少に応答してインスタンスをシャットダウンする操作を scale-in とよぶ． クラスターのスケーリングは， API サーバーではもちろんのこと，データベースサーバー・ストレージサーバーでも必要になる． ストレージサーバーでは，例えば頻繁にアクセスされるデータはキャッシュ領域に保存したり，データのコピーを複数作るなどのスケーリングが行われる． データベースサーバーも同様に，頻繁にアクセスされるデータのアクセスがパンクしてしまわないよう，分散的な処理が必要となる． このように，クラウドシステム内すべての箇所で，負荷が均一になるような調整が必要であり，開発者は多くの時間をそのチューニングに費やさなければならない． また，サービスの利用者の数などに応じてスケーリングの設定は常に見直される必要があり，継続的な開発が要求される．

さらに問題を複雑にするのは，API サーバーで処理されるべきタスクが，非一様な点である． 非一様であるとは，たとえばタスク A は 3000 ミリ秒の実行時間と 512MB のメモリーを消費し，別のタスク B は 1000 ミリ秒の実行時間と 128MB のメモリーを消費する，というような状況を指している． 一つのサーバーマシンが計算負荷が異なる複数のタスクを処理する場合，クラスターのスケーリングはより複雑になる． この状況をシンプルにするために，１サーバーで実行するタスクは１種類に限る，という設計も可能であるが，そうするとで生まれる弊害も多い (ほとんど使われないタスクに対してもサーバー一台をまるまる割り当てなければならない = ほとんどアイドリング状態になってしまう，など)．

Serverful なクラウドシステム

Serverless クラウドへ ​

Serverful クラウド (従来型) で議論したように，クラスターのスケーリングはクラウドシステムの経済的効率とシステムの安定性を最大化するために必須の作業である． それを反映して，多くの開発者の時間が投資されてきた．

クラスターのスケーリングはすべての開発者が何度も繰り返し行ってきた作業であり，いくつかの側面をテンプレート化し，共通化することができたならば開発のコストを大幅に削減できるだろう． それを実現するには，根本的なレベルからクラウドシステムの設計を考え直す必要がある． スケーリングを前提として考えることで，もっとシンプルで見通しがよいクラウドシステムの設計の仕組みはないだろうか？ そのような動機が，サーバーレスアーキテクチャが誕生する背後にあった．

従来の serverful なシステムでの最大の問題点は，サーバーをまるまる占有してしまうという点にある． すなわち， EC2 インスタンスを起動したとき，そのインスタンスは起動したユーザーだけが使えるものであり，計算のリソース (CPU や RAM) が独占的に割り当てられた状態になる． 固定した計算資源の割り当てがされてしまっているので，インスタンスの計算負荷が 0%であろうが 100%であろうが，均一の使用料金が起動時間に比例して発生する．

サーバーレスアーキテクチャは，このような 独占的に割り当てられた計算リソースというものを完全に廃止することを出発点とする． サーバーレスアーキテクチャでは，計算のリソースは，クラウドプロバイダーがすべて管理する． クライアントは，仮想インスタンスを一台まるごと借りるのではなく，計算のタスクの需要が生まれる毎に，実行したいプログラム・コマンドをクラウドに提出する． クラウドプロバイダーは，自身のもつ巨大な計算リソースから空きを探し，提出されたプログラムを実行し，実行結果をクライアントに返す． 言い換えると，計算リソースのスケーリングやアロケーションなどはクラウドプロバイダーが一手に引き受け，ユーザーはジョブをサブミットすることに注力する，という枠組みである． これを図示すると， figure_title のようになる．

従来のクラウドと Serverless クラウドの比較

サーバーレスクラウドでは，スケーリングはすべてクラウドプロバイダーが引き受けるので，スケーラビリティーが保証されている． クライアントが同時に大量のタスクを送信した場合でも，クラウドプロバイダー側の独自の仕組みによってすべてのタスクが遅延なく実行される． また，サーバーレスクラウドを利用することで，クラウドのコストは実際に使用した計算の総量 (稼働時間) で決定されることになる． これは，計算の実行総量に関わらずインスタンスの起動時間で料金が決定されていた従来のシステムと比べて大きな違いである．

サーバーレスクラウドは，従来のクラウドとは根本から異なったアプローチなので，コードの書き方やシステムの設計が大きく異なる． サーバーレスクラウドを開発・運用するには，サーバーレス固有の概念や用語に精通している必要がある． 以降では，実際にクラウドを動かしながら，サーバーレスをより具体的に体験していこう．

従来型の(仮想インスタンスをたくさん起動するような)クラウドシステムは，賃貸と似ているかもしれない． 部屋を借りるというのは，その部屋でどれだけの時間を過ごそうが，月々の家賃は一定である． 同様に，仮想サーバーも，それがどれほどの計算を行っているかに関わらず，一定の料金が時間ごとに発生する．

一方で，サーバーレスクラウドは，電気・水道・ガス料金 と似ている． こちらは，実際に使用した量に比例して料金が決定されている． サーバーレスクラウドも，実際に計算を行った総時間で料金が決まる仕組みになっている．

サーバーレスクラウドを構成するコンポーネント ​

サーバーレスアーキテクチャの概要がわかってきたところで，ここでは AWS においてサーバーレスクラウドを構成する様々なコンポーネントを紹介していこう． 特に， Lambda, S3, DynamoDB を取り上げ，解説する (figure_title)． サーバーレスクラウドは，これらのコンポーネントを統合することで一つのシステムが出来上がる． ここでは， Lambda，S3，DynamoDB を利用する際に押さえておかなければならない知識を一通り説明しきる都合上，具体的なイメージがわきにくいかもしれない． が，続く ??? でそれぞれについてハンズオン形式で演習を行うので，そこでさらに理解を深めれば大丈夫である．

Lambda, S3, DynamoDB のアイコン

Lambda ​

AWS でサーバーレスコンピューティングの中心を担うのが， Lambda である． Lambda の使い方を figure_title に図示している． Lambda の仕組みはシンプルで，まずユーザーは実行したいプログラムのコードを事前に登録しておく． プログラムは， Python, Node.js, Ruby などの主要な言語がサポートされている． Lambda に登録されたひとつひとつのプログラムを関数 (Function) とよぶ． そして，関数を実行したいときに，invoke コマンドを Lambda に送信する． Lambda では， invoke のリクエストを受け取るとただちに (数ミリセカンドから数百ミリセカンド程度の時間で) プログラムの実行を開始する． そして，実行結果をクライアントやその他の計算機に返す．

AWS Lambda

このように，Lambda では占有された仮想インスタンスは存在せず，実行を待っているプログラムだけがある状態である． invoke のリクエストに応じて，プログラムが AWS の巨大な計算機プールのどこかに配置され，実行される． 同時に複数のリクエストが来た場合でも， AWS はそれらを実行するための計算リソースを割り当て，並列的に処理を行ってくれる． 原理上は，数千から数万のリクエストが同時に来たとしても， Lambda はそれらを同時に実行することができる． このような，占有された仮想サーバーの存在なしに，動的に関数を実行するサービスのことを総称して FaaS (Function as a Service) とよぶ．

Lambda ではそれぞれの関数につき， 128MB から 10240MB のメモリーを使用することができる (執筆時点の仕様)． また，実効的な CPU のパワーはメモリーの量に比例する形で割り当てられる． すなわち，タスクに割り当てたメモリーの量が多ければ多いほど，より多くの CPU リソースが割り当てられることになる (しかし， RAM と CPU パワーの具体的な換算表は AWS からは公開されていない)． 実行時間は 100 ミリ秒の単位で記録され，実行時間に比例して料金が決定される． table_title は Lambda の利用料金表である (執筆時点で ap-north-east1 リージョンを選択した場合)．

Lambda の料金表
Memory (MB)Price per 100ms

128

$0.0000002083

512

$0.0000008333

1024

$0.0000016667

3008

$0.0000048958

実行時間に比例する料金に追加して，リクエストを送信するごとに発生する料金が設定されている． これは，百万回のリクエストにつき $0.2 である． たとえば， 128MB のメモリーを使用する関数を，それぞれ 200 ミリ秒，合計で 100 万回実行した場合， 0.0000002083 * 2 * 10^6 + 0.2 = $0.6 の料金となる． ウェブサーバーのデータベースの更新など簡単な計算であれば，200 ミリ秒程度で実行できる関数も多いことから，100 万回データベースの更新を行ったとしても，たった $0.6 しかコストが発生しないことになる． また，コードが実行されず待機状態になっている場合は，発生する料金は 0 である． このように，実際に意味のある処理が行われた時間にのみ，料金が発生する仕組みになっている．

Lambda は比較的短時間で完了する，反復性の高いタスクの実行に向いている． データベースの読み書きはその典型的な例であるが，そのほかにも，画像のサイズをトリミングしたり，サーバーサイドで定期的に実行されるメンテナンス処理などの利用が考えられる． また，複数の Lambda をリレー式に繋げることも可能で，シンプルな処理を組み合わせることで複雑なロジックを表現することができる．

上述の Lambda の料金計算は，説明のためコストに寄与する要素をいくつか省いている点は承知いただきたい． 例えば， DynamoDB の読み書きに関する料金や，ネットワークの通信にかかわるコストが考慮されていない．

サーバーレスストレージ: S3 ​

サーバーレスの概念は，ストレージにも拡張されている．

従来的なストレージ (ファイルシステム) では，必ずホストとなるマシンと OS が存在しなければならない． したがって，それほどパワーは必要ないまでも，ある程度の CPU リソースを割かなければならない． また，従来的なファイルシステムでは，データ領域のサイズは最初にディスクを初期化するときに決めなければならず，後から容量を増加させることはしばしば困難である (ZFS などのファイルシステムを使えばある程度は自由にファイルシステムのサイズを変更することは可能である)． よって，従来的なクラウドでは，ストレージを借りる際にはあらかじめディスクのサイズを指定せねばならず，ディスクの中身が空であろうと満杯であろうと，同じ利用料金が発生することになる (figure_title)．

Simple Storage Service (S3) は，サーバーレスなストレージシステムを提供する (figure_title)． S3 は従来的なストレージシステムと異なり， OS に"マウントする”という概念はない． 基本的に API を通じてデータの読み書きの操作が行われる． また，データの冗長化や暗号化，バックアップの作成など，通常ならば OS と CPU が介在しなければならない操作も， API を通じて行うことができる． S3 では事前に決められたディスク領域のサイズはなく，データを入れれば入れた分だけ，保存領域は拡大していく (仕様上はペタバイトスケールのデータを保存することが可能である)． ストレージにかかる料金は，保存してあるデータの総容量で決定される．

S3 と従来的なファイルシステムの比較

S3 を利用する際に，料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 説明のため主要な事項のみ取り出している． 詳細は 公式ドキュメンテーション "Amazon S3 pricing" を参照)．

S3 の利用料金
項目料金

Data storage (First 50TB)

$0.023 per GB per month

PUT, COPY, POST, LIST requests (per 1,000 requests)

$0.005

GET, SELECT, and all other requests (per 1,000 requests)

$0.0004

Data Transfer IN To Amazon S3 From Internet

$0

Data Transfer OUT From Amazon S3 To Internet

$0.09 per GB

第一に，データの保存には $0.025 per GB のコストが月ごとに発生する． したがって，1000GB のデータを S3 に一ヵ月保存した場合， $25 の料金が発生することになる． また，PUT, COPY, POST などのリクエスト (=データを書き込む操作) に対しては，データ容量に関係なく，1000 回ごとに $0.005 のコストが発生する． GET, SELECT などのリクエスト (=データを読み込む操作) に対しては，1000 回ごとに $0.0004 のコストが発生する． また， S3 はデータを外に取り出す際の通信にもコストが生じる． 執筆時点では，S3 からインターネットを通じて外部にデータを転送 (data-out) すると $0.09 per GB のコストが発生する． データをインターネットを通じて S3 に入れる (data-in) 通信は無料で行える． また， AWS の 同じ Region 内のサービス (Lambda や EC2 など) にデータを転送するのは無料である． AWS のリージョンをまたいだデータの転送にはコストが発生する． いずれにせよ，サーバーレスの概念に則り，すべての料金が従量課金制で決定される設定になっている．

サーバーレスデータベース: DynamoDB ​

サーバーレスの概念は，データベースにも適用することができる．

ここでいうデータベースとは， Web サービスなどにおけるユーザーや商品の情報を記録しておくための保存領域のことを指している． 従来的に有名なデータベースとしては MySQL, PostgreSQL, MongoDB などが挙げられる． データベースと普通のストレージの違いは，データの検索機能にある． 普通のストレージではデータは単純にディスクに書き込まれるだけだが， データベースでは検索がより効率的になるようなデータの配置がされたり， 頻繁にアクセスされるデータはメモリーにキャッシュされるなどの機能が備わっている． これにより，巨大なデータの中から，興味のある要素を高速に取得することができる．

このような検索機能を実現するには，当然 CPU の存在が必須である． したがって，従来的なデータベースを構築する際は，ストレージ領域に加えて，たくさんの CPU コアを搭載したマシンが用いられることが多い． また，データベースが巨大な場合は複数マシンにまたがった分散型のシステムが設計される． 分散型システムの場合は， Serverful クラウド (従来型) で議論したようにデータベースへのアクセス負荷に応じて適切なスケーリングがなされる必要がある．

DynamoDB は， AWS が提供しているサーバーレスな分散型データベースである． サーバーレスであるので，占有されたデータベース用仮想インスタンスは存在せず， API を通じてデータの書き込み・読み出し・検索などの操作を行う． S3 と同様に，データ保存領域の上限は定められておらず，データを入れれば入れた分だけ，保存領域は拡大していく． また，データベースへの負荷が増減したときのスケーリングは， DynamoDB が自動で行うので，ユーザーは心配する必要はない．

DynamoDB での利用料金の計算はやや複雑なのだが， "On-demand Capacity" というモードで使用した場合の料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 詳細は 公式ドキュメンテーション "Pricing for On-Demand Capacity" を参照)．

DynamoDB の利用料金
項目料金

Write request units

$1.25 per million write request units

Read request units

$0.25 per million read request units

Data storage

$0.25 per GB-month

DynamoDB ではデータの書き込み操作の単位を write request unit とよび，データの読み込み操作の単位を read request unit とよぶ． 基本的に， 1kB 以下のデータを一度書き込むと 1 write request unit を消費し，4kB 以下のデータを一度読み込むと 1 read request unit を消費する (詳しくは 公式ドキュメンテーション "Read/Write Capacity Mode" を参照のこと)． write request units は 100 万回ごとに $1.25, read request units は 100 万回ごとに $0.25 のコストが設定されている． また，保存されたデータ容量に対して $0.25 per GB のコストが月ごとに発生する． DynamoDB は高速な検索機能などを備えたデータベースであるので， GB あたりのストレージコストは S3 に比べ 10 倍程度高い． DynamoDB のデータの転送に関わるコストは，同じリージョン内ならば data-in，data-out ともに $0 である． リージョンをまたいだ通信には別途コストが発生する．

その他のサーバーレスクラウドの構成要素 ​

以上で紹介した Lambda, S3, DynamoDB がサーバーレスクラウドの中で最も使用する頻度が高いサービスになる． その他のサーバーレスクラウドの構成要素を以下に列挙する． いくつかについては，今後のハンズオンを行う中で改めて解説を行う．

  • API Gateway: API を構築する際のルーティングを担う． ??? で取り上げる．

  • Fargate: ??? で触れた Fargate も，サーバーレスクラウドの要素の一部である． Lambda との違いは，Lambda よりも大容量のメモリーや CPU を要するような計算などを行うことができる点が挙げられる．

  • Simple Notification Service (SNS): サーバーレスのサービス間でイベントをやり取りするためのサービス．

  • Step Functions: サーバーレスのサービス間のオーケストレーションを担う．

サーバーレスアーキテクチャは万能か？

この問いへの答えは，筆者は NO であると考える．

ここまで，サーバーレスの利点を強調して説明をしてきたが，まだまだ新しい技術なだけに，欠点，あるいはサーバーフルなシステムに劣る点は数多くある．

大きな欠点を一つあげるとすれば，サーバーレスのシステムは各クラウドプラットフォームに固有なものなので，特定のプラットフォームでしか運用できないシステムになってしまう点であろう． AWS で作成したサーバーレスのシステムを， Google のクラウドに移植するには，かなり大掛かりなプログラムの書き換えが必要になる． 一方， serverful なシステムであれば，プラットフォーム間のマイグレーションは比較的簡単に行うことができる． クラウドプロバイダーとしては，自社のシステムへの依存度を強めることで，顧客を離さないようにするという狙いがあるのだろう…

その他，サーバーレスコンピューティングの欠点や今後の課題などは，次の論文で詳しく議論されている． 興味のある読者はぜひ読んでいただきたい．

- +
Toshiki's Note

Appearance

TOC

Serverless architecture ​

サーバーレスアーキテクチャ (Serverless architecture) あるいは サーバーレスコンピューティング (Serverless computing) とは，従来とは全く異なるアプローチに基づくクラウドシステムの設計方法である． 歴史的には， AWS が 2014 年に発表した Lamba がサーバーレスアーキテクチャの先駆けとされている． その後， Google や Microsoft などのクラウドプラットフォームも同様の機能の提供を開始している． サーバーレスアーキテクチャの利点は，スケーラブルなクラウドシステムを安価かつ簡易に作成できる点であり，近年いたるところで導入が進んでいる．

Serverless とは，文字どおりの意味としてはサーバーなしで計算をするということになるが，それは一体どういう意味だろうか？ サーバーレスについて説明するためには，まずは従来的な， "serverful" とよばれるようなシステムについて解説しなければならない．

Serverful クラウド (従来型) ​

従来的なクラウドシステムのスケッチを figure_title に示す． クライアントから送信されたリクエストは，最初に API サーバーに送られる． API サーバーでは，リクエストの内容に応じてタスクが実行される． タスクには，API サーバーだけで完結できるものもあるが，多くの場合，データベースの読み書きが必要である． データベースには，データベース専用の独立したサーバーマシンが用いられることが一般的である． また，画像や動画などの容量の大きいデータは，また別のストレージサーバーに保存されることが多い． これらの API サーバー，データベースサーバー，ストレージサーバーはそれぞれ独立したサーバーマシンであり， AWS の言葉では EC2 による仮想インスタンスを想定してもらったらよい．

多くのウェブサービスでは，多数のクライアントからのリクエストを処理するため，複数のサーバーマシンがクラウド内で起動し，負荷を分散するような設計がなされている． クライアントから来たリクエストを計算容量に余裕のあるサーバーに振り分けるような操作を Load balancing とよび，そのような操作を担当するマシンのことを Load balancer という．

計算負荷を分散する目的で多数のインスタンスを起動するのはよいのだが，計算負荷が小さすぎてアイドリング状態にあるようではコストと電力の無駄遣いである． したがって，すべてのサーバーが常に目標とする計算負荷を維持するよう，計算の負荷に応じてクラスター内の仮想サーバーの数を動的に増減させるような仕組みが必要である． そのような仕組みをクラスターのスケーリングとよび，負荷の増大に応答して新しい仮想インスタンスをクラスターに追加する操作を scale-out，負荷の減少に応答してインスタンスをシャットダウンする操作を scale-in とよぶ． クラスターのスケーリングは， API サーバーではもちろんのこと，データベースサーバー・ストレージサーバーでも必要になる． ストレージサーバーでは，例えば頻繁にアクセスされるデータはキャッシュ領域に保存したり，データのコピーを複数作るなどのスケーリングが行われる． データベースサーバーも同様に，頻繁にアクセスされるデータのアクセスがパンクしてしまわないよう，分散的な処理が必要となる． このように，クラウドシステム内すべての箇所で，負荷が均一になるような調整が必要であり，開発者は多くの時間をそのチューニングに費やさなければならない． また，サービスの利用者の数などに応じてスケーリングの設定は常に見直される必要があり，継続的な開発が要求される．

さらに問題を複雑にするのは，API サーバーで処理されるべきタスクが，非一様な点である． 非一様であるとは，たとえばタスク A は 3000 ミリ秒の実行時間と 512MB のメモリーを消費し，別のタスク B は 1000 ミリ秒の実行時間と 128MB のメモリーを消費する，というような状況を指している． 一つのサーバーマシンが計算負荷が異なる複数のタスクを処理する場合，クラスターのスケーリングはより複雑になる． この状況をシンプルにするために，１サーバーで実行するタスクは１種類に限る，という設計も可能であるが，そうするとで生まれる弊害も多い (ほとんど使われないタスクに対してもサーバー一台をまるまる割り当てなければならない = ほとんどアイドリング状態になってしまう，など)．

Serverful なクラウドシステム

Serverless クラウドへ ​

Serverful クラウド (従来型) で議論したように，クラスターのスケーリングはクラウドシステムの経済的効率とシステムの安定性を最大化するために必須の作業である． それを反映して，多くの開発者の時間が投資されてきた．

クラスターのスケーリングはすべての開発者が何度も繰り返し行ってきた作業であり，いくつかの側面をテンプレート化し，共通化することができたならば開発のコストを大幅に削減できるだろう． それを実現するには，根本的なレベルからクラウドシステムの設計を考え直す必要がある． スケーリングを前提として考えることで，もっとシンプルで見通しがよいクラウドシステムの設計の仕組みはないだろうか？ そのような動機が，サーバーレスアーキテクチャが誕生する背後にあった．

従来の serverful なシステムでの最大の問題点は，サーバーをまるまる占有してしまうという点にある． すなわち， EC2 インスタンスを起動したとき，そのインスタンスは起動したユーザーだけが使えるものであり，計算のリソース (CPU や RAM) が独占的に割り当てられた状態になる． 固定した計算資源の割り当てがされてしまっているので，インスタンスの計算負荷が 0%であろうが 100%であろうが，均一の使用料金が起動時間に比例して発生する．

サーバーレスアーキテクチャは，このような 独占的に割り当てられた計算リソースというものを完全に廃止することを出発点とする． サーバーレスアーキテクチャでは，計算のリソースは，クラウドプロバイダーがすべて管理する． クライアントは，仮想インスタンスを一台まるごと借りるのではなく，計算のタスクの需要が生まれる毎に，実行したいプログラム・コマンドをクラウドに提出する． クラウドプロバイダーは，自身のもつ巨大な計算リソースから空きを探し，提出されたプログラムを実行し，実行結果をクライアントに返す． 言い換えると，計算リソースのスケーリングやアロケーションなどはクラウドプロバイダーが一手に引き受け，ユーザーはジョブをサブミットすることに注力する，という枠組みである． これを図示すると， figure_title のようになる．

従来のクラウドと Serverless クラウドの比較

サーバーレスクラウドでは，スケーリングはすべてクラウドプロバイダーが引き受けるので，スケーラビリティーが保証されている． クライアントが同時に大量のタスクを送信した場合でも，クラウドプロバイダー側の独自の仕組みによってすべてのタスクが遅延なく実行される． また，サーバーレスクラウドを利用することで，クラウドのコストは実際に使用した計算の総量 (稼働時間) で決定されることになる． これは，計算の実行総量に関わらずインスタンスの起動時間で料金が決定されていた従来のシステムと比べて大きな違いである．

サーバーレスクラウドは，従来のクラウドとは根本から異なったアプローチなので，コードの書き方やシステムの設計が大きく異なる． サーバーレスクラウドを開発・運用するには，サーバーレス固有の概念や用語に精通している必要がある． 以降では，実際にクラウドを動かしながら，サーバーレスをより具体的に体験していこう．

従来型の(仮想インスタンスをたくさん起動するような)クラウドシステムは，賃貸と似ているかもしれない． 部屋を借りるというのは，その部屋でどれだけの時間を過ごそうが，月々の家賃は一定である． 同様に，仮想サーバーも，それがどれほどの計算を行っているかに関わらず，一定の料金が時間ごとに発生する．

一方で，サーバーレスクラウドは，電気・水道・ガス料金 と似ている． こちらは，実際に使用した量に比例して料金が決定されている． サーバーレスクラウドも，実際に計算を行った総時間で料金が決まる仕組みになっている．

サーバーレスクラウドを構成するコンポーネント ​

サーバーレスアーキテクチャの概要がわかってきたところで，ここでは AWS においてサーバーレスクラウドを構成する様々なコンポーネントを紹介していこう． 特に， Lambda, S3, DynamoDB を取り上げ，解説する (figure_title)． サーバーレスクラウドは，これらのコンポーネントを統合することで一つのシステムが出来上がる． ここでは， Lambda，S3，DynamoDB を利用する際に押さえておかなければならない知識を一通り説明しきる都合上，具体的なイメージがわきにくいかもしれない． が，続く ??? でそれぞれについてハンズオン形式で演習を行うので，そこでさらに理解を深めれば大丈夫である．

Lambda, S3, DynamoDB のアイコン

Lambda ​

AWS でサーバーレスコンピューティングの中心を担うのが， Lambda である． Lambda の使い方を figure_title に図示している． Lambda の仕組みはシンプルで，まずユーザーは実行したいプログラムのコードを事前に登録しておく． プログラムは， Python, Node.js, Ruby などの主要な言語がサポートされている． Lambda に登録されたひとつひとつのプログラムを関数 (Function) とよぶ． そして，関数を実行したいときに，invoke コマンドを Lambda に送信する． Lambda では， invoke のリクエストを受け取るとただちに (数ミリセカンドから数百ミリセカンド程度の時間で) プログラムの実行を開始する． そして，実行結果をクライアントやその他の計算機に返す．

AWS Lambda

このように，Lambda では占有された仮想インスタンスは存在せず，実行を待っているプログラムだけがある状態である． invoke のリクエストに応じて，プログラムが AWS の巨大な計算機プールのどこかに配置され，実行される． 同時に複数のリクエストが来た場合でも， AWS はそれらを実行するための計算リソースを割り当て，並列的に処理を行ってくれる． 原理上は，数千から数万のリクエストが同時に来たとしても， Lambda はそれらを同時に実行することができる． このような，占有された仮想サーバーの存在なしに，動的に関数を実行するサービスのことを総称して FaaS (Function as a Service) とよぶ．

Lambda ではそれぞれの関数につき， 128MB から 10240MB のメモリーを使用することができる (執筆時点の仕様)． また，実効的な CPU のパワーはメモリーの量に比例する形で割り当てられる． すなわち，タスクに割り当てたメモリーの量が多ければ多いほど，より多くの CPU リソースが割り当てられることになる (しかし， RAM と CPU パワーの具体的な換算表は AWS からは公開されていない)． 実行時間は 100 ミリ秒の単位で記録され，実行時間に比例して料金が決定される． table_title は Lambda の利用料金表である (執筆時点で ap-north-east1 リージョンを選択した場合)．

Lambda の料金表
Memory (MB)Price per 100ms

128

$0.0000002083

512

$0.0000008333

1024

$0.0000016667

3008

$0.0000048958

実行時間に比例する料金に追加して，リクエストを送信するごとに発生する料金が設定されている． これは，百万回のリクエストにつき $0.2 である． たとえば， 128MB のメモリーを使用する関数を，それぞれ 200 ミリ秒，合計で 100 万回実行した場合， 0.0000002083 * 2 * 10^6 + 0.2 = $0.6 の料金となる． ウェブサーバーのデータベースの更新など簡単な計算であれば，200 ミリ秒程度で実行できる関数も多いことから，100 万回データベースの更新を行ったとしても，たった $0.6 しかコストが発生しないことになる． また，コードが実行されず待機状態になっている場合は，発生する料金は 0 である． このように，実際に意味のある処理が行われた時間にのみ，料金が発生する仕組みになっている．

Lambda は比較的短時間で完了する，反復性の高いタスクの実行に向いている． データベースの読み書きはその典型的な例であるが，そのほかにも，画像のサイズをトリミングしたり，サーバーサイドで定期的に実行されるメンテナンス処理などの利用が考えられる． また，複数の Lambda をリレー式に繋げることも可能で，シンプルな処理を組み合わせることで複雑なロジックを表現することができる．

上述の Lambda の料金計算は，説明のためコストに寄与する要素をいくつか省いている点は承知いただきたい． 例えば， DynamoDB の読み書きに関する料金や，ネットワークの通信にかかわるコストが考慮されていない．

サーバーレスストレージ: S3 ​

サーバーレスの概念は，ストレージにも拡張されている．

従来的なストレージ (ファイルシステム) では，必ずホストとなるマシンと OS が存在しなければならない． したがって，それほどパワーは必要ないまでも，ある程度の CPU リソースを割かなければならない． また，従来的なファイルシステムでは，データ領域のサイズは最初にディスクを初期化するときに決めなければならず，後から容量を増加させることはしばしば困難である (ZFS などのファイルシステムを使えばある程度は自由にファイルシステムのサイズを変更することは可能である)． よって，従来的なクラウドでは，ストレージを借りる際にはあらかじめディスクのサイズを指定せねばならず，ディスクの中身が空であろうと満杯であろうと，同じ利用料金が発生することになる (figure_title)．

Simple Storage Service (S3) は，サーバーレスなストレージシステムを提供する (figure_title)． S3 は従来的なストレージシステムと異なり， OS に"マウントする”という概念はない． 基本的に API を通じてデータの読み書きの操作が行われる． また，データの冗長化や暗号化，バックアップの作成など，通常ならば OS と CPU が介在しなければならない操作も， API を通じて行うことができる． S3 では事前に決められたディスク領域のサイズはなく，データを入れれば入れた分だけ，保存領域は拡大していく (仕様上はペタバイトスケールのデータを保存することが可能である)． ストレージにかかる料金は，保存してあるデータの総容量で決定される．

S3 と従来的なファイルシステムの比較

S3 を利用する際に，料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 説明のため主要な事項のみ取り出している． 詳細は 公式ドキュメンテーション "Amazon S3 pricing" を参照)．

S3 の利用料金
項目料金

Data storage (First 50TB)

$0.023 per GB per month

PUT, COPY, POST, LIST requests (per 1,000 requests)

$0.005

GET, SELECT, and all other requests (per 1,000 requests)

$0.0004

Data Transfer IN To Amazon S3 From Internet

$0

Data Transfer OUT From Amazon S3 To Internet

$0.09 per GB

第一に，データの保存には $0.025 per GB のコストが月ごとに発生する． したがって，1000GB のデータを S3 に一ヵ月保存した場合， $25 の料金が発生することになる． また，PUT, COPY, POST などのリクエスト (=データを書き込む操作) に対しては，データ容量に関係なく，1000 回ごとに $0.005 のコストが発生する． GET, SELECT などのリクエスト (=データを読み込む操作) に対しては，1000 回ごとに $0.0004 のコストが発生する． また， S3 はデータを外に取り出す際の通信にもコストが生じる． 執筆時点では，S3 からインターネットを通じて外部にデータを転送 (data-out) すると $0.09 per GB のコストが発生する． データをインターネットを通じて S3 に入れる (data-in) 通信は無料で行える． また， AWS の 同じ Region 内のサービス (Lambda や EC2 など) にデータを転送するのは無料である． AWS のリージョンをまたいだデータの転送にはコストが発生する． いずれにせよ，サーバーレスの概念に則り，すべての料金が従量課金制で決定される設定になっている．

サーバーレスデータベース: DynamoDB ​

サーバーレスの概念は，データベースにも適用することができる．

ここでいうデータベースとは， Web サービスなどにおけるユーザーや商品の情報を記録しておくための保存領域のことを指している． 従来的に有名なデータベースとしては MySQL, PostgreSQL, MongoDB などが挙げられる． データベースと普通のストレージの違いは，データの検索機能にある． 普通のストレージではデータは単純にディスクに書き込まれるだけだが， データベースでは検索がより効率的になるようなデータの配置がされたり， 頻繁にアクセスされるデータはメモリーにキャッシュされるなどの機能が備わっている． これにより，巨大なデータの中から，興味のある要素を高速に取得することができる．

このような検索機能を実現するには，当然 CPU の存在が必須である． したがって，従来的なデータベースを構築する際は，ストレージ領域に加えて，たくさんの CPU コアを搭載したマシンが用いられることが多い． また，データベースが巨大な場合は複数マシンにまたがった分散型のシステムが設計される． 分散型システムの場合は， Serverful クラウド (従来型) で議論したようにデータベースへのアクセス負荷に応じて適切なスケーリングがなされる必要がある．

DynamoDB は， AWS が提供しているサーバーレスな分散型データベースである． サーバーレスであるので，占有されたデータベース用仮想インスタンスは存在せず， API を通じてデータの書き込み・読み出し・検索などの操作を行う． S3 と同様に，データ保存領域の上限は定められておらず，データを入れれば入れた分だけ，保存領域は拡大していく． また，データベースへの負荷が増減したときのスケーリングは， DynamoDB が自動で行うので，ユーザーは心配する必要はない．

DynamoDB での利用料金の計算はやや複雑なのだが， "On-demand Capacity" というモードで使用した場合の料金に関わってくる主要な事項をまとめたのが table_title である (us-east-1 リージョンのもの． 詳細は 公式ドキュメンテーション "Pricing for On-Demand Capacity" を参照)．

DynamoDB の利用料金
項目料金

Write request units

$1.25 per million write request units

Read request units

$0.25 per million read request units

Data storage

$0.25 per GB-month

DynamoDB ではデータの書き込み操作の単位を write request unit とよび，データの読み込み操作の単位を read request unit とよぶ． 基本的に， 1kB 以下のデータを一度書き込むと 1 write request unit を消費し，4kB 以下のデータを一度読み込むと 1 read request unit を消費する (詳しくは 公式ドキュメンテーション "Read/Write Capacity Mode" を参照のこと)． write request units は 100 万回ごとに $1.25, read request units は 100 万回ごとに $0.25 のコストが設定されている． また，保存されたデータ容量に対して $0.25 per GB のコストが月ごとに発生する． DynamoDB は高速な検索機能などを備えたデータベースであるので， GB あたりのストレージコストは S3 に比べ 10 倍程度高い． DynamoDB のデータの転送に関わるコストは，同じリージョン内ならば data-in，data-out ともに $0 である． リージョンをまたいだ通信には別途コストが発生する．

その他のサーバーレスクラウドの構成要素 ​

以上で紹介した Lambda, S3, DynamoDB がサーバーレスクラウドの中で最も使用する頻度が高いサービスになる． その他のサーバーレスクラウドの構成要素を以下に列挙する． いくつかについては，今後のハンズオンを行う中で改めて解説を行う．

  • API Gateway: API を構築する際のルーティングを担う． ??? で取り上げる．

  • Fargate: ??? で触れた Fargate も，サーバーレスクラウドの要素の一部である． Lambda との違いは，Lambda よりも大容量のメモリーや CPU を要するような計算などを行うことができる点が挙げられる．

  • Simple Notification Service (SNS): サーバーレスのサービス間でイベントをやり取りするためのサービス．

  • Step Functions: サーバーレスのサービス間のオーケストレーションを担う．

サーバーレスアーキテクチャは万能か？

この問いへの答えは，筆者は NO であると考える．

ここまで，サーバーレスの利点を強調して説明をしてきたが，まだまだ新しい技術なだけに，欠点，あるいはサーバーフルなシステムに劣る点は数多くある．

大きな欠点を一つあげるとすれば，サーバーレスのシステムは各クラウドプラットフォームに固有なものなので，特定のプラットフォームでしか運用できないシステムになってしまう点であろう． AWS で作成したサーバーレスのシステムを， Google のクラウドに移植するには，かなり大掛かりなプログラムの書き換えが必要になる． 一方， serverful なシステムであれば，プラットフォーム間のマイグレーションは比較的簡単に行うことができる． クラウドプロバイダーとしては，自社のシステムへの依存度を強めることで，顧客を離さないようにするという狙いがあるのだろう…

その他，サーバーレスコンピューティングの欠点や今後の課題などは，次の論文で詳しく議論されている． 興味のある読者はぜひ読んでいただきたい．

+ \ No newline at end of file diff --git a/development/aws/webserver.html b/development/aws/webserver.html index 4be3c1c3..13e9baa3 100644 --- a/development/aws/webserver.html +++ b/development/aws/webserver.html @@ -5,13 +5,13 @@ Web サービスの作り方 | Toshiki's Note - + - + @@ -33,9 +33,9 @@ -
Toshiki's Note

Appearance

TOC

Web サービスの作り方 ​

ここからが，本書第三部の内容になる． これまでのセクションでは，仮想サーバーをクラウド上に起動し，そこで計算を走らせる方法について解説をしてきた． EC2, ECS, Fargate, Batch などを利用して，動的にスケールするクラスターを構成し，並列にタスクを実行するクラウドシステムを実装してきた． 振り返ると，これまで紹介してきた内容は，自分自身が行いたい計算をクラウドを駆使することで実現する，という用途にフォーカスしていたことに気がつくだろう． 一方で，広く一般の人々に使ってもらえるような計算サービス・データベースを提供する，というのもクラウドの重要な役割として挙げられる．

本章から始まる第三部では，前回までとは少し方向性を変え，どのようにしてクラウド上にアプリケーションを展開し，広く一般の人に使ってもらうか，という点を講義したいと思う． これを通じて，どのようにして世の中のウェブサービスができ上がっているのかを知り，さらにどうやって自分でそのようなアプリケーションをゼロから構築するのか，という点を学んでもらう． その過程で，サーバーレスアーキテクチャという最新のクラウド設計手法を解説する．

その前準備として，本章ではどのようにしてウェブサービスが出来上がっているのか，その背後にある技術の概要を解説する． 用語の解説が中心となるが，後のハンズオンを実装するために必須の知識であるので，理解して前に進むよう心がけよう．

ウェブサービスの仕組み  — Twitter を例に ​

あなたがパソコンやスマートフォンから Twitter, Facebook, YouTube などのウェブサービスにアクセスしたとき，実際にどのようなことが行われ，コンテンツが提示されているのだろうか？

HTTP を通じたサーバーとクライアントのデータのやり取りは，すでに知っている読者も多いだろうし，逆にすべて解説しようとすると紙面が足りないので，ここではエッセンスの説明のみにとどめる． 以降では Twitter を具体例として，背後にあるサーバーとクライアントの間の通信を概説しよう． 概念図としては figure_title のような通信がクライアントとサーバーの間で行われていることになる．

クライアントと Web サーバーの通信の概念図

前提として，クライアントとサーバーの通信は HTTP (Hypertext Transfer Protocol) を使って行われる． また，最近では，暗号化された HTTP である HTTPS (HTTPS (Hypertext Transfer Protocol Secure)) を用いることがスタンダードになってきている． 第一のステップとして，クライアントは HTTP(S) 通信によってサーバーから静的なコンテンツを取得する． 静的なコンテンツとは， HTML (Hyptertext Markup Language) で記述されたウェブページの文書本体， CSS (Cascading Style Sheets) で記述されたページのデザインやレイアウトファイル，そして JavaScript (JS) で記述されたページの動的な挙動を定義したプログラム，が含まれる． Twitter を含む現代的なウェブアプリケーションの設計では，この静的なファイル群はページの”枠”を定義するだけで，中身となるコンテンツ (例: ツイートの一覧) は別途 API (Application Programming Interface) によって取得されなければならない． そこで，クライアントは先のステップで取得された JavaScript で定義されたプログラムに従って，サーバーに API を送信し，ツイートや画像データを取得する． この際，テキストデータのやり取りには JSON (JavaScript Object Notation) というフォーマットが用いられることが多い． 画像や動画などのコンテンツも同様に API により取得される． このようにして取得されたテキストや画像が，HTML の文書に埋め込まれることで，最終的にユーザーに提示されるページが完成するのである． また，新しいツイートを投稿するときにも，クライアントから API を通じてサーバーのデータベースにデータが書き込まれる．

REST API ​

API (Application Programming Interface) とはこれまで何度も出てきた言葉であるが，ここではよりフォーマルな定義付けを行う． API とはあるソフトウェア・アプリケーションが，外部のソフトウェアに対してコマンドやデータをやり取りするための媒介の一般的総称である． とくに，ウェブサービスの文脈では，サーバーが外界に対して提示しているコマンドの一覧のことを意味する． クライアントは，提示されている API から適切なコマンドを使うことによって，所望のデータを取得したり，あるいはサーバーにデータを送信したりする．

とくに，ウェブの文脈では REST (Representational State Transfer) とよばれる設計思想に基づいた API が現在では最も一般的に使われている． REST の設計指針に従った API のことを REST API あるいは RESTful API とよんだりする．

REST API は， figure_title に示したような Method と URI (Universal Resource Identifier) の組からなる．

REST API

Method (メソッド) とは，どのような操作を行いたいかを抽象的に表す，"動詞" として捉えることができる． メソッドには HTTP 規格で定義された 9 個の動詞 (verb) を使用することができる． この中でも， GET, POST, PUT, PATCH, DELETE の 5 個が最も頻繁に使用される (table_title)． この 5 つのメソッドによる操作を総称して CRUD (create, read, update, and delete) とよぶ．

REST API Methods
メソッド意図される動作

GET

要素を取得する

POST

新しい要素を作成する

PUT

既存の要素を新しい要素と置き換える

PATCH

既存の要素の一部を更新する

DELETE

要素を削除する

一方， URI は操作が行われる対象，すなわち "目的語" を表す． ウェブの文脈では操作が行われる対象のことをしばしば リソース とよぶ． URI は多くの場合 http または https から始まるウェブサーバーのアドレスから始まり， / (スラッシュ) 以降に所望のリソースのパスが指定される． figure_title の例で言えば， https://api.twitter.com というアドレスの /1.1/status/home_timeline というリソースを取得 (GET) せよ，という意味になる (なお，ここで 1.1 という数字は API のバージョンを示している)． この API リクエストによって，ユーザーのホームのタイムラインのツイートの一覧が取得される．

REST API のメソッドには， table_title で挙げたもの以外に， HTTP プロトコルで定義されているほかのメソッド (OPTIONS, TRACE など) を用いることもできるが，あまり一般的ではない．

また，これらのメソッドだけでは動詞として表現しきれないこともあるが， URI の名前でより意味を明確にすることもある． メソッドの使い方も，要素を削除する際は必ず DELETE を使わなければならない，という決まりもなく，たとえば， Twitter API でツイートを消す API は POST statuses/destroy/:id で定義されている． 最終的には，各ウェブサービスが公開している API ドキュメンテーションを読んで，それぞれの API がどんな操作をするのかを調べる必要がある．

REST の概念は 2000 年代初頭に確立され，今日の API 設計のスタンダードとなった． 一方で，ウェブのテクノロジーが進歩するにつれて，新たな API の設計アプローチの需要も高まっている． 近年とくに人気を集めているのが， GraphQL と呼ばれる API の設計方法である． GraphQL は Facebook 社によって最初に作られ，現在は GraghQL Foundation によって維持と更新がされている． GraphQL を使用すると，クライアントは REST と比較してより柔軟性の高いデータのクエリを行うことができるなど，いくつかの利点がある． キーワードだけでも知っておくと，今後役に立つだろう．

Twitter API ​

もう少し具体的にウェブサービスの API を体験する目的で，ここでは Twitter の API を見てみよう． Twitter が提供している API の一覧は Twitter の Developer Documentation で見ることができる． いくつかの代表的な API を table_title にまとめた．

Twitter API
エンドポイント動作

GET statuses/home_timeline

ホームのタイムラインのツイートの一覧を取得する．

GET statuses/show/:id

:id で指定されたツイートの詳細情報を取得する．

GET search

ツイートの検索を実行する．

POST statuses/update

新しいツイートを投稿する．

POST media/upload

画像をアップロードする

POST statuses/destroy/:id

:id で指定されたツイートを削除する．

POST statuses/retweet/:id

:id で指定されたツイートをリツイートする．

POST statuses/unretweet/:id

:id で指定されたツイートのリツイートを取り消す．

POST favorites/create

選択したツイートを"いいね"する．

POST favorites/destroy

選択したツイートを"いいね"を取り消す．

この API リストをもとに， Twitter のアプリまたはウェブサイトを開いたときに起こるクライアントとサーバーの通信をシミュレートしてみよう．

ユーザーが Twitter を開くと，まず最初に GET statuses/home_timeline の API リクエストによって，ユーザーのホームのタイムラインのツイートのリストが取得される． 個々のツイートは JSON 形式のデータになっており， id, text, user, coordinates, entities などの属性を含む． id はツイートに固有な ID を表し， text はツイートの本文を含んでいる． user はツイートを投稿したユーザーの名前やプロフィール画像の URL などを含んだ JSON データになっている． coordinates にはツイートが発信された地理的な座標が記録されている． また， entities にはツイートに関連するメディアファイル (画像など) のリンクなどの情報が埋め込まれている． GET statuses/home_timeline からは直近のツイートのリスト (リストが長すぎる場合は途中で切られたもの) が取得される． もしツイートの ID を知っている場合は GET statuses/show/:id を呼ぶことによって， :id パラメータで指定された特定のツイートを取得することができる．

ツイートの検索を行うためには GET search API を使用する． この API には，ツイートに含まれる単語や，ハッシュタグ，ツイートの発信された日時や場所など，様々なクエリの条件を渡すことができる． API からは， GET statuses/home_timeline などと同様， JSON 形式のツイートのデータが返される．

ユーザーが新しいツイートを投稿するには POST statuses/update のエンドポイントを利用する． POST statuses/update には，ツイートの文章や，リプライの場合はリプライ先のツイートの ID などのデータを送信する． また，ツイートに画像データを添付したい場合は， POST media/upload を併せて使用する． ツイートの削除を行うには， POST statuses/destroy/:id を用いる．

そのほか，頻繁に行われる操作としては， POST statuses/retweet/:id と POST statuses/unretweet/:id がある． これらは， :id で指定されるツイートに対して，それぞれリツイートを実行あるいは取り消すための API である． また， POST favorites/create，POST favorites/destroy を使用することによって，選択されたツイートに"いいね"を追加したり，取り消したりする操作を行う．

このような一連の操作が， Twitter のアプリの背後では行われている． また，自分自身でボットを作成したい場合は，これらの API を適切に組み合わせ，カスタムのプログラムを書くことで実現される．

このように， API はあらゆるウェブサービスを作るうえで一番基礎となる要素である． 次からの章では本章で紹介した用語が何度も出てくるので，頭の片隅に置いたうえで読み進めていただきたい．

- +
Toshiki's Note

Appearance

TOC

Web サービスの作り方 ​

ここからが，本書第三部の内容になる． これまでのセクションでは，仮想サーバーをクラウド上に起動し，そこで計算を走らせる方法について解説をしてきた． EC2, ECS, Fargate, Batch などを利用して，動的にスケールするクラスターを構成し，並列にタスクを実行するクラウドシステムを実装してきた． 振り返ると，これまで紹介してきた内容は，自分自身が行いたい計算をクラウドを駆使することで実現する，という用途にフォーカスしていたことに気がつくだろう． 一方で，広く一般の人々に使ってもらえるような計算サービス・データベースを提供する，というのもクラウドの重要な役割として挙げられる．

本章から始まる第三部では，前回までとは少し方向性を変え，どのようにしてクラウド上にアプリケーションを展開し，広く一般の人に使ってもらうか，という点を講義したいと思う． これを通じて，どのようにして世の中のウェブサービスができ上がっているのかを知り，さらにどうやって自分でそのようなアプリケーションをゼロから構築するのか，という点を学んでもらう． その過程で，サーバーレスアーキテクチャという最新のクラウド設計手法を解説する．

その前準備として，本章ではどのようにしてウェブサービスが出来上がっているのか，その背後にある技術の概要を解説する． 用語の解説が中心となるが，後のハンズオンを実装するために必須の知識であるので，理解して前に進むよう心がけよう．

ウェブサービスの仕組み  — Twitter を例に ​

あなたがパソコンやスマートフォンから Twitter, Facebook, YouTube などのウェブサービスにアクセスしたとき，実際にどのようなことが行われ，コンテンツが提示されているのだろうか？

HTTP を通じたサーバーとクライアントのデータのやり取りは，すでに知っている読者も多いだろうし，逆にすべて解説しようとすると紙面が足りないので，ここではエッセンスの説明のみにとどめる． 以降では Twitter を具体例として，背後にあるサーバーとクライアントの間の通信を概説しよう． 概念図としては figure_title のような通信がクライアントとサーバーの間で行われていることになる．

クライアントと Web サーバーの通信の概念図

前提として，クライアントとサーバーの通信は HTTP (Hypertext Transfer Protocol) を使って行われる． また，最近では，暗号化された HTTP である HTTPS (HTTPS (Hypertext Transfer Protocol Secure)) を用いることがスタンダードになってきている． 第一のステップとして，クライアントは HTTP(S) 通信によってサーバーから静的なコンテンツを取得する． 静的なコンテンツとは， HTML (Hyptertext Markup Language) で記述されたウェブページの文書本体， CSS (Cascading Style Sheets) で記述されたページのデザインやレイアウトファイル，そして JavaScript (JS) で記述されたページの動的な挙動を定義したプログラム，が含まれる． Twitter を含む現代的なウェブアプリケーションの設計では，この静的なファイル群はページの”枠”を定義するだけで，中身となるコンテンツ (例: ツイートの一覧) は別途 API (Application Programming Interface) によって取得されなければならない． そこで，クライアントは先のステップで取得された JavaScript で定義されたプログラムに従って，サーバーに API を送信し，ツイートや画像データを取得する． この際，テキストデータのやり取りには JSON (JavaScript Object Notation) というフォーマットが用いられることが多い． 画像や動画などのコンテンツも同様に API により取得される． このようにして取得されたテキストや画像が，HTML の文書に埋め込まれることで，最終的にユーザーに提示されるページが完成するのである． また，新しいツイートを投稿するときにも，クライアントから API を通じてサーバーのデータベースにデータが書き込まれる．

REST API ​

API (Application Programming Interface) とはこれまで何度も出てきた言葉であるが，ここではよりフォーマルな定義付けを行う． API とはあるソフトウェア・アプリケーションが，外部のソフトウェアに対してコマンドやデータをやり取りするための媒介の一般的総称である． とくに，ウェブサービスの文脈では，サーバーが外界に対して提示しているコマンドの一覧のことを意味する． クライアントは，提示されている API から適切なコマンドを使うことによって，所望のデータを取得したり，あるいはサーバーにデータを送信したりする．

とくに，ウェブの文脈では REST (Representational State Transfer) とよばれる設計思想に基づいた API が現在では最も一般的に使われている． REST の設計指針に従った API のことを REST API あるいは RESTful API とよんだりする．

REST API は， figure_title に示したような Method と URI (Universal Resource Identifier) の組からなる．

REST API

Method (メソッド) とは，どのような操作を行いたいかを抽象的に表す，"動詞" として捉えることができる． メソッドには HTTP 規格で定義された 9 個の動詞 (verb) を使用することができる． この中でも， GET, POST, PUT, PATCH, DELETE の 5 個が最も頻繁に使用される (table_title)． この 5 つのメソッドによる操作を総称して CRUD (create, read, update, and delete) とよぶ．

REST API Methods
メソッド意図される動作

GET

要素を取得する

POST

新しい要素を作成する

PUT

既存の要素を新しい要素と置き換える

PATCH

既存の要素の一部を更新する

DELETE

要素を削除する

一方， URI は操作が行われる対象，すなわち "目的語" を表す． ウェブの文脈では操作が行われる対象のことをしばしば リソース とよぶ． URI は多くの場合 http または https から始まるウェブサーバーのアドレスから始まり， / (スラッシュ) 以降に所望のリソースのパスが指定される． figure_title の例で言えば， https://api.twitter.com というアドレスの /1.1/status/home_timeline というリソースを取得 (GET) せよ，という意味になる (なお，ここで 1.1 という数字は API のバージョンを示している)． この API リクエストによって，ユーザーのホームのタイムラインのツイートの一覧が取得される．

REST API のメソッドには， table_title で挙げたもの以外に， HTTP プロトコルで定義されているほかのメソッド (OPTIONS, TRACE など) を用いることもできるが，あまり一般的ではない．

また，これらのメソッドだけでは動詞として表現しきれないこともあるが， URI の名前でより意味を明確にすることもある． メソッドの使い方も，要素を削除する際は必ず DELETE を使わなければならない，という決まりもなく，たとえば， Twitter API でツイートを消す API は POST statuses/destroy/:id で定義されている． 最終的には，各ウェブサービスが公開している API ドキュメンテーションを読んで，それぞれの API がどんな操作をするのかを調べる必要がある．

REST の概念は 2000 年代初頭に確立され，今日の API 設計のスタンダードとなった． 一方で，ウェブのテクノロジーが進歩するにつれて，新たな API の設計アプローチの需要も高まっている． 近年とくに人気を集めているのが， GraphQL と呼ばれる API の設計方法である． GraphQL は Facebook 社によって最初に作られ，現在は GraghQL Foundation によって維持と更新がされている． GraphQL を使用すると，クライアントは REST と比較してより柔軟性の高いデータのクエリを行うことができるなど，いくつかの利点がある． キーワードだけでも知っておくと，今後役に立つだろう．

Twitter API ​

もう少し具体的にウェブサービスの API を体験する目的で，ここでは Twitter の API を見てみよう． Twitter が提供している API の一覧は Twitter の Developer Documentation で見ることができる． いくつかの代表的な API を table_title にまとめた．

Twitter API
エンドポイント動作

GET statuses/home_timeline

ホームのタイムラインのツイートの一覧を取得する．

GET statuses/show/:id

:id で指定されたツイートの詳細情報を取得する．

GET search

ツイートの検索を実行する．

POST statuses/update

新しいツイートを投稿する．

POST media/upload

画像をアップロードする

POST statuses/destroy/:id

:id で指定されたツイートを削除する．

POST statuses/retweet/:id

:id で指定されたツイートをリツイートする．

POST statuses/unretweet/:id

:id で指定されたツイートのリツイートを取り消す．

POST favorites/create

選択したツイートを"いいね"する．

POST favorites/destroy

選択したツイートを"いいね"を取り消す．

この API リストをもとに， Twitter のアプリまたはウェブサイトを開いたときに起こるクライアントとサーバーの通信をシミュレートしてみよう．

ユーザーが Twitter を開くと，まず最初に GET statuses/home_timeline の API リクエストによって，ユーザーのホームのタイムラインのツイートのリストが取得される． 個々のツイートは JSON 形式のデータになっており， id, text, user, coordinates, entities などの属性を含む． id はツイートに固有な ID を表し， text はツイートの本文を含んでいる． user はツイートを投稿したユーザーの名前やプロフィール画像の URL などを含んだ JSON データになっている． coordinates にはツイートが発信された地理的な座標が記録されている． また， entities にはツイートに関連するメディアファイル (画像など) のリンクなどの情報が埋め込まれている． GET statuses/home_timeline からは直近のツイートのリスト (リストが長すぎる場合は途中で切られたもの) が取得される． もしツイートの ID を知っている場合は GET statuses/show/:id を呼ぶことによって， :id パラメータで指定された特定のツイートを取得することができる．

ツイートの検索を行うためには GET search API を使用する． この API には，ツイートに含まれる単語や，ハッシュタグ，ツイートの発信された日時や場所など，様々なクエリの条件を渡すことができる． API からは， GET statuses/home_timeline などと同様， JSON 形式のツイートのデータが返される．

ユーザーが新しいツイートを投稿するには POST statuses/update のエンドポイントを利用する． POST statuses/update には，ツイートの文章や，リプライの場合はリプライ先のツイートの ID などのデータを送信する． また，ツイートに画像データを添付したい場合は， POST media/upload を併せて使用する． ツイートの削除を行うには， POST statuses/destroy/:id を用いる．

そのほか，頻繁に行われる操作としては， POST statuses/retweet/:id と POST statuses/unretweet/:id がある． これらは， :id で指定されるツイートに対して，それぞれリツイートを実行あるいは取り消すための API である． また， POST favorites/create，POST favorites/destroy を使用することによって，選択されたツイートに"いいね"を追加したり，取り消したりする操作を行う．

このような一連の操作が， Twitter のアプリの背後では行われている． また，自分自身でボットを作成したい場合は，これらの API を適切に組み合わせ，カスタムのプログラムを書くことで実現される．

このように， API はあらゆるウェブサービスを作るうえで一番基礎となる要素である． 次からの章では本章で紹介した用語が何度も出てくるので，頭の片隅に置いたうえで読み進めていただきたい．

+ \ No newline at end of file diff --git a/development/file-naming-convention.html b/development/file-naming-convention.html new file mode 100644 index 00000000..de0885d2 --- /dev/null +++ b/development/file-naming-convention.html @@ -0,0 +1,106 @@ + + + + + + File Name Conventions | Toshiki's Note + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Toshiki's Note

Appearance

TOC

File Name Conventions ​

A collection of guidelines for writing file names used in web projects.

Possible characters ​

Use dashes as delimiters ​

  • You should use dashes (-) as delimiters.
  • Periods are allowed in some cases, such as for languages and conditions.
  • Never use spaces or underscores. Spaces are converted to %20 in URLs or can break an URL when shared. Underscores are difficult to see when the file name is displayed as an underlined link. Although the use of underscores does not impact your ranking that much, Google advices not to use underscores.
  • Exceptions apply for functional requirements, such as for Sass partials. A leading underscore informs the Sass compiler a file is only a partial file and should never be generated into a stand alone CSS file.

Right:

file-name-with-dashes.en.min.html
file-name-with-dashes.en.min.html

Do not use special characters ​

Avoid using non-alphanumeric characters in file names, such as: '*' ':' '\' '/' '<' '>' '|' '"' '!' '?' '[' ']' ';' '=' '+' '&' '£' '$' '€' '%' or ','. These characters can have special meaning in programming languages or can cause problems with different operating systems.

Use lowercase, never uppercase ​

We should always consider URLs as case-sensitive according to W3.org. Therefore, use lowercase to reduces errors when typing URLs.

Write sections of a file name in a consistent order ​

Sections should be written in this order:

  1. Description
  2. Number
  3. Date
  4. Target device, image size, pixel density
  5. Version number
  6. Status
  7. Language code
  8. File conditions

Possible combinations

description-01-20150102-palm-1.0b-draft.en.min.js /* extreme combination */
+description.min.js
+description.en.html
+description-01.jpg
+description-02.jpg
+description-1024x768_2x.jpg
+description-desk_2x.jpg
description-01-20150102-palm-1.0b-draft.en.min.js /* extreme combination */
+description.min.js
+description.en.html
+description-01.jpg
+description-02.jpg
+description-1024x768_2x.jpg
+description-desk_2x.jpg

Write description for developers and users ​

Don't be afraid to write long informative file names. Few people type a file name manually and most operating systems support 255 characters. But only add information that makes it easy for users and developers to recognize files from one another at a glance. For the description use information such as:

  • type of data
  • project name or acronym
  • subjects
  • people's names
  • characteristics
  • location

Give your images detailed, informative filenames The filename can give Google clues about the subject matter of the image. Try to make your filename a good description of the subject matter of the image. For example, my-new-black-kitten.jpg is a lot more informative than IMG00023.JPG. Descriptive filenames can also be useful to users: If we're unable to find suitable text in the page on which we found the image, we'll use the filename as the image's snippet in our search results. - Google, 2015, Image publishing guidelines

Keep people's names compact ​

Sometimes you want to include the name of a person in the file name, e.g. the author or the person in the photo.

  • Write names without word delimiters.
  • Only write initials for the first name and write the last name in full. There are two exceptions: (1) when the last name or final file name is very long you may use initials for the last name; (2) when there is already a person with the same combination, you may write the first name in full.
  • If people have exactly the same name written in full, you can add a number: firstnamelastname2.

Right:

bvandebiezen.jpg
+shoogenhout.jpg
+a-very-long-description-with-name-bvdb.jpg
+asmith.jpg
+adamsmith.jpg
+adamsmith2.jpg
bvandebiezen.jpg
+shoogenhout.jpg
+a-very-long-description-with-name-bvdb.jpg
+asmith.jpg
+adamsmith.jpg
+adamsmith2.jpg

Use two or more digits to distinguish sequential files with the same description ​

  • Start number with a dash as a delimiter.
  • Add a zero to single digit numbers, e.g. '01' instead of '1'.
  • Numbers may also be placed before the description if needed. A dash will still be used as delimiter with the description.

Right:

description-01.jpg
+description-02.jpg
+description-03.jpg
+description-04.jpg
+
+01-description.jpg
+02-description.jpg
+03-description.jpg
+04-description.jpg
description-01.jpg
+description-02.jpg
+description-03.jpg
+description-04.jpg
+
+01-description.jpg
+02-description.jpg
+03-description.jpg
+04-description.jpg

Keep dates or date ranges compact and start with year ​

  • Write dates without delimiters.
  • Always use four digits for years, two digits for months and two digits for days.
  • Start with year, then month, and end with day if available: yyyymmdd, yyyymm, or yyyy. This makes sure similar file names are sorted by date when sorted alphabetically.
  • Use a double dash to separate two dates describing an interval: yyyy--yyyy. Start with the earliest date.

Right:

description-20150401.php
+description-201504.php
+description-2015.php
+description-2000--2010.php
description-20150401.php
+description-201504.php
+description-2015.php
+description-2000--2010.php

See 'ISO 8601' for further reading.

Use special modifiers for target devices, image sizes or media queries, and pixel densities. ​

Modifiers are inspired by Apple iOS naming conventions. There are some differences. Apple uses '@' as a delimiter for the section indicating higher resolution images, for example '@2x' for retina images. Because '@' is a reserved character and can create problems, we use Bourbon's convention: an underscore. Also, Apple uses a tilde (~) as a delimiter for a section indicating specific devices. Because also a tilde can create problems, we suggest to simply use a dash.

  • Order should be: (1) target device or media query, (2) size, (3) pixel density.
  • Start target device or media queries with a dash (-) as delimiter.
  • Start image sizes with a dash (-) as delimiter.
  • Start pixel density with an underscore (_) as delimiter, for example '_2x' or '_3x'.
  • When only a width or height is available or applicable, add a 'w' for width or 'h' for height directly after the the amount of pixels.
  • When both measurements are available, do not add a 'w' or 'h' and separate the width and height with an 'x'.
  • When both the width and height should not exceed a dimension but the images should keep the original aspect ratio, add a 'max' (maximum) after the amount of pixels.

Right:

description_2x.jpg
+description-lap.jpg
+description-desk.jpg
+description-lap_2x.jpg
+description-palm-1024w_2x.jpg
+description-iphone5-568h_2x.jpg
+description-palm-1024x768_2x.jpg
+description-40max.jpg
description_2x.jpg
+description-lap.jpg
+description-desk.jpg
+description-lap_2x.jpg
+description-palm-1024w_2x.jpg
+description-iphone5-568h_2x.jpg
+description-palm-1024x768_2x.jpg
+description-40max.jpg

Use version numbers if available ​

  • Start version with a dash (-) as delimiter.
  • Use periods (.) to separate point releases.
  • Always add trailing zeros to major releases, e.g. '2.0' instead of '2'.
  • Types, such as 'a' (alpha), 'b' (beta), 'rc1' (release candidate 1) can be added without delimiters.

Right:

description-0.5.js
+description-1.0b.js
+description-1.0rc1.js
description-0.5.js
+description-1.0b.js
+description-1.0rc1.js

Add status when needed ​

  • You can optionally add a file status such as 'draft' and 'published'.
  • Start status with a dash.

Right:

description-draft.md
description-draft.md

Add language code only when different languages are available ​

  • Use a period to separate the language code from the rest of the file name.
  • Use ISO 639-1, two-letter codes, for languages.
  • Only add languages when different languages are available.

Right:

description.nl.txt
+description.en.txt
description.nl.txt
+description.en.txt

Add file conditions just before the file extension ​

  • The file condition should be the last part, just before the file extension.
  • Use a period (.) to separate the condition from the rest of the file name.
  • Use periods (.) as a delimiter for different conditions.

Right:

description.min.js
+description.custom1234.min.js
description.min.js
+description.custom1234.min.js

Rewrite original file names not following conventions ​

It is not preferred to keep file names in it's original format if it doesn't match your file name conventions. But in some cases it is easier to keep the file name untouched. Sometimes you want to easily replace a file with a newer one from the original source in the future.

+ + + + \ No newline at end of file diff --git a/development/rclone-for-r2.html b/development/rclone-for-r2.html new file mode 100644 index 00000000..4e331544 --- /dev/null +++ b/development/rclone-for-r2.html @@ -0,0 +1,94 @@ + + + + + + Configure rclone for R2 | Toshiki's Note + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Toshiki's Note

Appearance

TOC

Configure rclone for R2 ​

Example of how to configure rclone to use R2.

You must generate an Access Key before getting started. All examples will utilize access_key_id and access_key_secret variables which represent the Access Key ID and Secret Access Key values you generated.

With [rclone](https://rclone.org/install/) installed, you may run [rclone config](https://rclone.org/s3/) to configure a new S3 storage provider. You will be prompted with a series of questions for the new prvider details.

If you have already configured rclone in the past, you may run rclone config file to print the location of your rclone configuration file:

bash
$ rclone config file
+# Configuration file is stored at:
+# ~/.config/rclone/rclone.conf
$ rclone config file
+# Configuration file is stored at:
+# ~/.config/rclone/rclone.conf

Then use an editor (nano or vim, for example) to add or edit the new provider. This example assumes you are adding a new r2demo provider:

bash
[r2demo]
+type = s3
+provider = Cloudflare
+access_key_id = abc123 # Your access_key_id
+secret_access_key = xyz456 # Your access_key_secret
+endpoint = https://<accountid>.r2.cloudflarestorage.com
+acl = private
[r2demo]
+type = s3
+provider = Cloudflare
+access_key_id = abc123 # Your access_key_id
+secret_access_key = xyz456 # Your access_key_secret
+endpoint = https://<accountid>.r2.cloudflarestorage.com
+acl = private

You may then use the new rclone provider for any of your normal workflows.

List buckets & objects ​

The rclone tree command can be used to list the contents of the remote, in this case Cloudflare R2.

bash
$ rclone tree r2demo:
+# /
+# ├── user-uploads
+# │   └── foobar.png
+# └── my-bucket-name
+#     ├── cat.png
+#     └── todos.txt
+
+$ rclone tree r2demo:my-bucket-name
+# /
+# ├── cat.png
+# └── todos.txt
$ rclone tree r2demo:
+# /
+# ├── user-uploads
+# │   └── foobar.png
+# └── my-bucket-name
+#     ├── cat.png
+#     └── todos.txt
+
+$ rclone tree r2demo:my-bucket-name
+# /
+# ├── cat.png
+# └── todos.txt

Upload and retrieve objects ​

The rclone copy command can be used to upload objects to an R2 bucket and vice versa - this allows you to upload files up to the 5 TB maximum object size that R2 supports.

bash
# Upload dog.txt to the user-uploads bucket
+$ rclone copy dog.txt r2demo:user-uploads/dog.txt
+$ rclone tree r2demo:user-uploads
+# /
+# ├── foobar.png
+# └── dog.txt
+
+# Download dog.txt from the user-uploads bucket
+$ rclone copy r2demo:user-uploads/dog.txt dog.txt
# Upload dog.txt to the user-uploads bucket
+$ rclone copy dog.txt r2demo:user-uploads/dog.txt
+$ rclone tree r2demo:user-uploads
+# /
+# ├── foobar.png
+# └── dog.txt
+
+# Download dog.txt from the user-uploads bucket
+$ rclone copy r2demo:user-uploads/dog.txt dog.txt

Generate presigned URLs ​

You can also generate presigned links which allow you to share public access to a file temporarily using the rclone link command.

+ + + + \ No newline at end of file diff --git a/getting-started.html b/getting-started.html index 0d9389a6..765d9d43 100644 --- a/getting-started.html +++ b/getting-started.html @@ -5,12 +5,12 @@ Git push results in "Authentication Failed” | Toshiki's Note - + - + @@ -42,9 +42,9 @@ remote: Invalid username or password. fatal: Authentication failed for 'https://github.com/andatoshiki/andatoshiki.git/'

git push to GitHub remote branch failed via https protocol

Solution ​

Solution 1 ​

If you enabled two-factor authentication in your GitHub account you won't be able to push via HTTPS using your accounts password. Instead you need to generate a personal access token. This can be done in the application settings of your GitHub account. Using this token as your password should allow you to push to your remote repository via HTTPS. Use your username as usual.

You may also need to update the origin for your repository if it is set to HTTPS. Do this to switch to SSH:

bash
$ git remote -v
 $ git remote set-url origin git@github.com:$USERNAME/$REPONAME.git
$ git remote -v
-$ git remote set-url origin git@github.com:$USERNAME/$REPONAME.git

Solution 2 ​

GitHub username & password/access token mismatched due to prior temporary Git config setup → Reset Git global username & email config or in current repository

To set your global username/email configuration:

  1. Open the command line.
  2. Set your username: git config --global user.name "FIRST_NAME LAST_NAME"
  3. Set your email address: git config --global user.email "MY_NAME@example.com"

To set repository-specific username/email configuration:

  1. From the command line, change into the repository directory.
  2. Set your username: git config user.name "FIRST_NAME LAST_NAME"
  3. Set your email address: git config user.email "MY_NAME@example.com"
  4. Verify your configuration by displaying your configuration file: cat .git/config
- +$ git remote set-url origin git@github.com:$USERNAME/$REPONAME.git

Solution 2 ​

GitHub username & password/access token mismatched due to prior temporary Git config setup → Reset Git global username & email config or in current repository

To set your global username/email configuration:

  1. Open the command line.
  2. Set your username: git config --global user.name "FIRST_NAME LAST_NAME"
  3. Set your email address: git config --global user.email "MY_NAME@example.com"

To set repository-specific username/email configuration:

  1. From the command line, change into the repository directory.
  2. Set your username: git config user.name "FIRST_NAME LAST_NAME"
  3. Set your email address: git config user.email "MY_NAME@example.com"
  4. Verify your configuration by displaying your configuration file: cat .git/config
+ \ No newline at end of file diff --git a/hashmap.json b/hashmap.json index b847ff56..9693dbc7 100644 --- a/hashmap.json +++ b/hashmap.json @@ -1 +1 @@ -{"academic_chemistry_index.md":"e22e1eca","academic_physics_index.md":"669c3eb1","academic_chemistry_notes_12-5.md":"fa35f18c","development_aws_scientific-computing.md":"6e6683eb","development_aws_serverless.md":"7d293d5a","development_aws_webserver.md":"ab41d017","getting-started.md":"bd1ab99e","index.md":"6f6f33ad","javascript_notes_1_1-1.md":"10b5b6ea","javascript_notes_1_1-2.md":"5a4dd0dd","roadmap.md":"7ba79349","save_reading_index.md":"78fc5e98","save_reading_outliers_1.md":"daf781a1","save_reading_outliers_3.md":"6e397ae9","save_reading_outliers_4.md":"a4a3d333","academic_chemistry_problems_03-02-2.md":"bab6348d","save_reading_outliers_2.md":"8febfe5f","academic_literature_writing_methods-of-development.md":"cb0d57b6","academic_chemistry_problems_03-02-1.md":"3f33f487","academic_physics_ipho-formulas-jpn_11.md":"3d1ed58e","academic_physics_ipho-formulas-jpn_6.md":"02f25efb","academic_physics_ipho-formulas-jpn_12.md":"6b0b8711","academic_physics_ipho-formulas-jpn_13.md":"b330f9cc","academic_physics_ipho-formulas-jpn_2.md":"3b1f7ddd","academic_chemistry_problems_02-20.md":"e1fa1931","development_aws_license.md":"48a0d402","academic_chemistry_problems_03-02-3.md":"90fcd861","academic_physics_ipho-formulas-jpn_10.md":"31925895","academic_physics_ipho-formulas-jpn_3.md":"417386cd","academic_vocabulary_2023_02_2023-02-27.md":"07288c48","academic_vocabulary_index.md":"d46af4f9","application_markdown-it-katex_how-to-use.md":"ae049342","application_markdown-it-katex_tips.md":"e64f6d16","development_aws_readme.md":"6ebeba49","development_aws_acknowledgement.md":"d9cfe93b","development_aws_appendix.md":"79df306e","development_aws_assignments.md":"3f42c3cd","development_aws_author.md":"09627572","development_aws_aws-batch.md":"83297645","development_aws_aws-get-started.md":"df995644","development_aws_closing.md":"b2d7e915","development_aws_cloud.md":"34efce50","development_aws_docker-system.md":"209ef2e6","development_aws_handson-bashoutter.md":"4863f098","development_aws_handson-ec2.md":"64c416d6","development_aws_handson-qabot.md":"2f7eac1b","development_aws_handson-serverless.md":"38596c90","development_aws_introduction.md":"222bee89","academic_literature_index.md":"e8ed8386","academic_physics_ipho-formulas-jpn_9.md":"6637313d","academic_physics_ipho-formulas-jpn_1.md":"ec1dbaf7","academic_physics_ipho-formulas-jpn_5.md":"d14a555b","academic_physics_ipho-formulas-jpn_7.md":"95da3e92","academic_physics_ipho-formulas-jpn_8.md":"544b0b43","academic_physics_ipho-formulas-jpn_4.md":"8aeca4b5","development_aws_handson-jupyter.md":"c0838dc1","development_aws_main.md":"dd2f6ba8","application_markdown-it-katex_support-function.md":"aeaaab84","application_markdown-it-katex_support-table.md":"ff72261b"} +{"academic_chemistry_index.md":"20e560a6","academic_chemistry_problems_03-02-3.md":"2d09c751","development_aws_handson-qabot.md":"b244b90e","development_aws_handson-serverless.md":"a47ced9b","development_aws_license.md":"3681aae7","academic_literature_index.md":"b9024d79","academic_literature_writing_methods-of-development.md":"b79948e6","academic_physics_index.md":"837a8425","academic_physics_ipho-formulas-jpn_3.md":"24fb7253","academic_chemistry_notes_12-5.md":"7fe55c03","academic_physics_ipho-formulas-jpn_12.md":"bff53a1c","academic_chemistry_problems_03-02-2.md":"35916f88","development_aws_handson-ec2.md":"d527a33c","development_aws_handson-jupyter.md":"887c5eff","academic_physics_ipho-formulas-jpn_5.md":"db50d685","academic_physics_ipho-formulas-jpn_8.md":"02aeb40e","academic_vocabulary_2023_02_2023-02-27.md":"50b28361","academic_vocabulary_index.md":"b0a1288c","application_markdown-it-katex_how-to-use.md":"b720171d","application_markdown-it-katex_tips.md":"2d4f69d7","development_aws_readme.md":"ea2d6a9b","development_aws_acknowledgement.md":"2c27a486","development_aws_appendix.md":"7272d46f","development_aws_assignments.md":"27524939","development_aws_author.md":"a50d2f4c","academic_physics_ipho-formulas-jpn_10.md":"53acc7b0","development_aws_handson-bashoutter.md":"91af8955","academic_physics_ipho-formulas-jpn_2.md":"2b6166b8","academic_chemistry_problems_03-02-1.md":"765a5cd1","academic_physics_ipho-formulas-jpn_6.md":"2e655de1","academic_physics_ipho-formulas-jpn_7.md":"7be7b512","development_aws_introduction.md":"a28e3444","academic_physics_ipho-formulas-jpn_4.md":"6ae94d2b","development_aws_aws-get-started.md":"305acfd1","development_aws_aws-batch.md":"2225dbd4","academic_physics_ipho-formulas-jpn_9.md":"fc474d28","development_aws_closing.md":"73173906","development_aws_cloud.md":"d6418ba0","development_aws_docker-system.md":"b4b42fe0","academic_physics_ipho-formulas-jpn_13.md":"40dc3c65","development_aws_scientific-computing.md":"44b26ae4","development_aws_serverless.md":"954dd7a1","development_aws_webserver.md":"a8b089c4","development_file-naming-convention.md":"404e82de","development_rclone-for-r2.md":"e7f3a9b7","academic_physics_ipho-formulas-jpn_11.md":"80185913","academic_chemistry_problems_02-20.md":"6c822dc9","getting-started.md":"ad0edf0e","index.md":"4559d5f1","javascript_notes_1_1-1.md":"7c957dab","javascript_notes_1_1-2.md":"46f1230f","roadmap.md":"a42ebae3","save_reading_index.md":"983fe09d","save_reading_outliers_1.md":"e3cd2c6d","save_reading_outliers_2.md":"ad1783d7","save_reading_outliers_3.md":"e6fcee78","save_reading_outliers_4.md":"62591abe","academic_physics_ipho-formulas-jpn_1.md":"1be1e0d1","development_aws_main.md":"cc9bca88","application_markdown-it-katex_support-function.md":"1d1d4129","application_markdown-it-katex_support-table.md":"646afbf1"} diff --git a/index.html b/index.html index 3300ae27..f0ac0c0c 100644 --- a/index.html +++ b/index.html @@ -3,14 +3,14 @@ - Toshiki's Notebook | Eternal digital knowledge base for content craetion and notes management. + Toshiki's Notebook | Eternal digital knowledge base for content creation and notes management. - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

Toshiki's Notebook

Research & Produce

👨‍💻 Eternal & digital knowledge base for content craetion and notes management.

👉 Start Reading
🗒️ View on GitHub
Home logo
🕒

Tempus Fugit

Tempus Fugit, which means "time flies" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential.

☕

Carpe Diem

Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning.

💡

Epiphania

An epiphany, derived from the Latin word "epiphania," is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas.

- +
Toshiki's Note

Appearance

Toshiki's Notebook

Research & Produce

👨‍💻 Eternal & digital knowledge base for content creation and notes management.

👉 Start Reading
🗒️ View on GitHub
Home logo
🕒

Tempus Fugit

Tempus Fugit, which means "time flies" in Latin, is a phrase that highlights the fact that every person has the same 24 hours per day to learn. However, this time is never enough to learn everything. That's why recording knowledge for review is essential.

☕

Carpe Diem

Carpe Diem, take it slow, seize the day and savor its moments. Enjoy a cup of coffee while playing blues on a 1980s-style CD player. Turn off the lights, close the curtains, and let the small lamp illuminate the space. The time is yours, relish the day, and unleash the productivity while learning.

💡

Epiphania

An epiphany, derived from the Latin word "epiphania," is a moment of sudden and brilliant realization or insight. These moments of clarity and inspiration are precious and should be treated as such; to ensure that we don't forget these valuable ideas.

+ \ No newline at end of file diff --git a/javascript/notes/1/1-1.html b/javascript/notes/1/1-1.html index 8c9f48fe..c33e3848 100644 --- a/javascript/notes/1/1-1.html +++ b/javascript/notes/1/1-1.html @@ -5,12 +5,12 @@ 1-1: Values | Toshiki's Note - + - + @@ -38,9 +38,9 @@ 'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'
'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'
 'Lorem ipsum dolor sit amet, consectetur adipiscing elit.'

TIP

Both single and double quotes can be used to mark strings

  • As long as the quotes at the start and the end of the string match.

Almost anything can be put between quotes, and JavaScript will make a string value out of it. But the followings are tricky to be put between quotes.

  • Newlines: The things we get when we press enter on keyboard. We uses \n to represent.
  • New tab: Similarly like newlines, \t indicates a new tab being indented.

Take the following string as an example,

js
'Lorem ipsum dolor sit amet,\n consectetur adipiscing elit.'
'Lorem ipsum dolor sit amet,\n consectetur adipiscing elit.'

The following will be the rendered output,

js
Lorem ipsum dolor sit amet,
 consectetur adipiscing elit.
Lorem ipsum dolor sit amet,
-consectetur adipiscing elit.

WARNING

But the situations where we want a backslash in a string to just be a backslash instead of a special code. By using \\ would render the "just slash" to \ on output, instead of a special character, as follows,

js
"A newline character is written like \"\\n\"."
"A newline character is written like \"\\n\"."

Strings cannot be divided, multiplied, or subtracted, unlike integers. But the + operator can be used on the. It concatenates (glues) the two strings together, the following example will produce the string "concatenate".

js
'con' + 'cat' + 'e' + 'nate'
'con' + 'cat' + 'e' + 'nate'
1-1-3: Strings review
  • Nearly everything could be put into strings in JavaScript
  • String could be wrapped with "" double quotation marks or '' single quotation marks.
  • There is a tricky situation in JavaScript's string, where \ backslash followed with a character are being introduced specially by the interpreter, such as \n referred as newline, \t referred as new tab.
    • \\ double backslash will be rendered as a single \.
    • \& symbols followed after slash will not contain its original function such as connecting, but will be rendered as & on output.

1-1-3-FR: Further Reading ​

From the example given in the prior section with a lot of backslashes, it might be hard to understand the concept on at what time the slash will be rendered and when it will be considered as a special character starter.

The example is given as,

js
"A newline character is written like \"\\n\"."
"A newline character is written like \"\\n\"."

The rendered output will be on print,

js
A newline character is written like "\n".
A newline character is written like "\n".

the backslash character (\) is used as an escape character to indicate that the following character(s) should be treated specially. In this case, the "\n" sequence is an escape sequence that represents a newline character. The backslash before the n character tells the JavaScript interpreter that it should treat the n as a special character and not just as the letter "n".

However, if we want to include an actual backslash character (\), we will have to escape it by using to two backslashes \\, or else with one single slash the JavaScript interpreter will still determines it as an special operator, Thus \\ will print \ on output.

  • But what about the quotation marks used within the quotation marks that wraps the string? Wouldn't JavaScript interpreter throw error?

    • In the given JavaScript string, the inner set of double quotation marks ("\\n") is escaped using a backslash (\) character. This tells the JavaScript interpreter to treat the inner double quotation marks as a regular character instead of a string delimiter.

      So when the string is rendered on output, the backslash character will be removed and the inner set of quotation marks will be displayed as a regular character. The rendered string will look like this,

      js
      '\n'
      '\n'

      The outer set of double quotation marks in the original string delimit the entire string and will be displayed as regular quotation marks.

Source: ChatGPT CA (Code Analysis)

- +consectetur adipiscing elit.

WARNING

But the situations where we want a backslash in a string to just be a backslash instead of a special code. By using \\ would render the "just slash" to \ on output, instead of a special character, as follows,

js
"A newline character is written like \"\\n\"."
"A newline character is written like \"\\n\"."

Strings cannot be divided, multiplied, or subtracted, unlike integers. But the + operator can be used on the. It concatenates (glues) the two strings together, the following example will produce the string "concatenate".

js
'con' + 'cat' + 'e' + 'nate'
'con' + 'cat' + 'e' + 'nate'
1-1-3: Strings review
  • Nearly everything could be put into strings in JavaScript
  • String could be wrapped with "" double quotation marks or '' single quotation marks.
  • There is a tricky situation in JavaScript's string, where \ backslash followed with a character are being introduced specially by the interpreter, such as \n referred as newline, \t referred as new tab.
    • \\ double backslash will be rendered as a single \.
    • \& symbols followed after slash will not contain its original function such as connecting, but will be rendered as & on output.

1-1-3-FR: Further Reading ​

From the example given in the prior section with a lot of backslashes, it might be hard to understand the concept on at what time the slash will be rendered and when it will be considered as a special character starter.

The example is given as,

js
"A newline character is written like \"\\n\"."
"A newline character is written like \"\\n\"."

The rendered output will be on print,

js
A newline character is written like "\n".
A newline character is written like "\n".

the backslash character (\) is used as an escape character to indicate that the following character(s) should be treated specially. In this case, the "\n" sequence is an escape sequence that represents a newline character. The backslash before the n character tells the JavaScript interpreter that it should treat the n as a special character and not just as the letter "n".

However, if we want to include an actual backslash character (\), we will have to escape it by using to two backslashes \\, or else with one single slash the JavaScript interpreter will still determines it as an special operator, Thus \\ will print \ on output.

  • But what about the quotation marks used within the quotation marks that wraps the string? Wouldn't JavaScript interpreter throw error?

    • In the given JavaScript string, the inner set of double quotation marks ("\\n") is escaped using a backslash (\) character. This tells the JavaScript interpreter to treat the inner double quotation marks as a regular character instead of a string delimiter.

      So when the string is rendered on output, the backslash character will be removed and the inner set of quotation marks will be displayed as a regular character. The rendered string will look like this,

      js
      '\n'
      '\n'

      The outer set of double quotation marks in the original string delimit the entire string and will be displayed as regular quotation marks.

Source: ChatGPT CA (Code Analysis)

+ \ No newline at end of file diff --git a/javascript/notes/1/1-2.html b/javascript/notes/1/1-2.html index 617d19d6..1f513335 100644 --- a/javascript/notes/1/1-2.html +++ b/javascript/notes/1/1-2.html @@ -5,12 +5,12 @@ 1-2 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC
- +
Toshiki's Note

Appearance

TOC
+ \ No newline at end of file diff --git a/roadmap.html b/roadmap.html index 1792980f..24456bb6 100644 --- a/roadmap.html +++ b/roadmap.html @@ -5,12 +5,12 @@ Theme | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Theme ​

Homepage ​

  • Finish three hero action blocks with keywords and its corresponded descriptions.

  • Finish my learning platform section using SVGs of different online academies from simple icons.

    • Descriptions

    • Icon in SVG format

    • Platform name

    • Linkable SVG icon/button

      • Khan academy
      • GitHub
      • Libretext
      • OpenStax
      • BCcampus
      • Wikibooks
      • Wikipedia
      • Bookboon
      • [ ]

Configs ​

  • Header configs including different properties and meta configs.
  • Complete navigation configuration.
  • Complete sidebar configs for different directories.
  • Vitepress sidebar generation plugin for huge bundled clips article directory.

Articles ​

  • <head> tag documentation or guide by joshbuchea/HEAD.
  • Write index page for every directory, no Chat GPT allowed.
  • Chemistry review notes.
  • JavaScript 0-100 notes per chapter.
    • Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)
  • Outliers novel clips
  • Migrate clipped articles and archives from Notion to Vitepress.
  • Fresh copy of "How To Ask Questions The Smart Way" by Eric Steven Raymond
- +
Toshiki's Note

Appearance

TOC

Theme ​

Homepage ​

  • Finish three hero action blocks with keywords and its corresponded descriptions.

  • Finish my learning platform section using SVGs of different online academies from simple icons.

    • Descriptions

    • Icon in SVG format

    • Platform name

    • Linkable SVG icon/button

      • Khan academy
      • GitHub
      • Libretext
      • OpenStax
      • BCcampus
      • Wikibooks
      • Wikipedia
      • Bookboon
      • [ ]

Configs ​

  • Header configs including different properties and meta configs.
  • Complete navigation configuration.
  • Complete sidebar configs for different directories.
  • Vitepress sidebar generation plugin for huge bundled clips article directory.

Articles ​

  • <head> tag documentation or guide by joshbuchea/HEAD.
  • Write index page for every directory, no Chat GPT allowed.
  • Chemistry review notes.
  • JavaScript 0-100 notes per chapter.
    • Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)
  • Outliers novel clips
  • Migrate clipped articles and archives from Notion to Vitepress.
  • Fresh copy of "How To Ask Questions The Smart Way" by Eric Steven Raymond
+ \ No newline at end of file diff --git a/save/reading/index.html b/save/reading/index.html index 05e835b3..c568d1bd 100644 --- a/save/reading/index.html +++ b/save/reading/index.html @@ -5,12 +5,12 @@ Welcome to Reading Notes | Toshiki's Note - + - + @@ -32,9 +32,9 @@ - - + + \ No newline at end of file diff --git a/save/reading/outliers/1.html b/save/reading/outliers/1.html index 6460b146..9734ec91 100644 --- a/save/reading/outliers/1.html +++ b/save/reading/outliers/1.html @@ -5,12 +5,12 @@ Introduction & Chapter 1: The Roseto Mystery | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Introduction & Chapter 1: The Roseto Mystery ​

Summary: Introduction: The Roseto Mystery ​

Section 1. ​

In the late 1800s, immigrants from Roseto Valfortore, Italy, came to Pennsylvania to work in the slate quarry near the town of Bangor. The settlers established a new community, named after their old one: Roseto. They built schools, a park, small shops, and more than a dozen factories. They kept to themselves and did not interact much with the predominantly German and English populations of the towns around them.

In the 1950s, a local doctor told Stewart Wolf, a medical researcher, that hardly anyone in Roseto under age sixty-five had heart disease. Wolf was curious. He did blood draws and EKG scans, and he analyzed physicians’ records and death certificates. Wolf was amazed that almost no one under fifty-five showed any signs of heart disease, which at the time was America’s leading cause of death in men under sixty-five. Wolf enlisted the help of John Bruhn, a sociologist. They found that there was no suicide, alcoholism, or drug addiction in Roseto: everyone seemed to be dying of old age. According to Gladwell, Roseto was an outlier.

Section 2. ​

Wolf was forced to rule out diet and lifestyle as explanations, because the Rosetans had poor eating habits, smoked heavily, and were obese. He ruled out genetics after determining that Rosetans’ relatives in other parts of the country were not in comparably good health. Wolf eventually decided that what made Roseto’s inhabitants so healthy was the town itself—its culture and social structure. Many homes contained three generations living together, and with a population of just two thousand people, there were twenty-two separate civic organizations.

Wolf and Bruhn were met with a great deal of skepticism. They had to work hard to convince the medical establishment that the values of where we live and the people around us have a significant impact on who we are. Gladwell wants Outliers to do for our understanding of success what Wolf did for our understanding of health.

Summary: Chapter One: The Matthew Effect ​

Section 1. ​

Gladwell describes a game between two teams in the Canadian Hockey League, the most competitive junior hockey league in the world. Starting at a very young age, players are regularly assessed for talent, so that the best can be separated out and prepared for the next level. The system is designed to be a meritocracy.

Section 2. ​

Gladwell wants to show that circumstances matter just as much for success as work ethic or intelligence. People often ask about the lifestyle, intelligence, or special talents of successful individuals, but not enough people ask when and where a person grew up. Gladwell likens people to trees: we all know that the tallest trees received the most sunlight, had nutrient-rich soil, and were lucky enough to avoid lumberjacks. He suggests that we study the environmental and social influences that make people successful.

Section 3. ​

In the 1980s, Canadian psychologist Roger Barnsley observed that on elite Canadian hockey teams, whether youth league or NHL, a disproportionate number of players have birthdays in the first three months of the year.

Section 4. ​

The explanation for the distribution of birthdays is simple: the cutoff date for an individual’s league assignment is January 1. A player born on January 2 will be 12 months older than one born at the end of December. At the age of nine or ten, this often translates into a huge difference in size, coordination, and maturity. By age ten, coaches regularly pick the oldest and largest potential players for the more competitive “rep squads,” where the players will receive better coaching, have more practice time, and play many more games. By thirteen or fourteen, all of the extra experience and coaching will have actually made the players better, and they will be selected for the more competitive league(s). The same thing happens in American youth baseball and European soccer, but with different cutoff dates.

The phenomenon is also observable in education. Economists Kelly Bedard and Elizabeth Dhuey found that on math and science tests, the older children in fourth grade regularly score better than the youngest children. This can affect whether a child qualifies for a gifted program. When Bedard and Dhuey extended their analysis to four-year colleges, they found that the students who belonged to the relatively youngest group were underrepresented by more than 10 percent.

Section 5. ​

The top-level hockey players were given opportunities they neither deserved nor earned. Sociologist Robert Merton called this the “Matthew Effect,” after the verse in the Gospel of Matthew:

“For unto everyone that hath shall be given, and he shall have abundance. But from that hath not shall be taken away even that which he hath.”

The successful receive more resources: the best students receive the best teaching, the richest citizens get the biggest tax breaks, the largest kids get the best coaching. Sociologists call this “accumulative advantage.” Gladwell points out that cutoff dates exclude nearly half of the potential athletes from a population. Junior hockey teams in the Czech Republic show the same unbalanced distribution of birthdays as in Canada. If Canada or the Czech Republic separated each of their youth leagues into two separate leagues, one for kids born in the first half of the year, and one for kids born in the second half of the year, they would have twice as many athletes available for their national teams. Similarly, schools could separate kindergarten classes into students born in the first, middle, and last four months of the year. The benefits would justify the added administrative work, but, writes Gladwell, the scheme would run afoul of society’s need to believe that individual merit matters more than the rules society has created.

Section 6. ​

Gord Wasden, the father of one of the Canadian Hockey League boys from the first section, describes his son. Scott Wasden worked hard to get where he is, but he has also generally had the advantage of being big compared to others on his team. He was born on January 4.

Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect ​

Gladwell's thesis argues that that the idea of rugged, individual success is not accurate. Rather, the most successful person doesn’t thrive without some environmental and social influence plus a dose of good fortune.  After laying out the concepts of a meritocracy and suggesting that this is the way that hockey players advance in Canada, he immediately undercuts this idea. Gladwell uses the illustration of birthday distribution as an example of an invisible contributor to the success of hockey players. With a registration cutoff date of January 1, players born from January 2 through March 31 are older than teammates born later in the year. These older players are likely bigger, faster, more coordinated, and more mature. Since the older players are seen as more skilled, they get selected for more advanced leagues with more practice, more games, and higher-quality coaching. By the time these older players reach their early teens, the extra advantages gained from being judged better because of their birthday translate into being better players. Similarly, the age cutoff for school registrations often mistakes maturity for ability. As a result, the older children appear smarter, get put into advanced groups, and qualify for gifted programs. The birthdate bias carries through to college.

Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that "success leads to more success." More simply: by being a little bit better, a hockey player will get opportunities that may result in the player becoming an outlier. Gladwell implies that the systems that determine success are not efficient. He asks readers to question the way we, as a society, think about success. He suggests that changing the systems in athletics and education would chip away at the myth of individual merit as the chief marker of success and help level the playing field. Gladwell's model for this phenomenon is hockey player Scott Wasden. By demonstrating that Scott had the passion, talent, and work ethic for the game, as well as the good luck to be born on January 4, Gladwell reasons that Wasden, like many others he will be profiling, benefit by being both good and lucky.

Reference

  • “Outliers Introduction & Chapter 1 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section1/.
- +
Toshiki's Note

Appearance

TOC

Introduction & Chapter 1: The Roseto Mystery ​

Summary: Introduction: The Roseto Mystery ​

Section 1. ​

In the late 1800s, immigrants from Roseto Valfortore, Italy, came to Pennsylvania to work in the slate quarry near the town of Bangor. The settlers established a new community, named after their old one: Roseto. They built schools, a park, small shops, and more than a dozen factories. They kept to themselves and did not interact much with the predominantly German and English populations of the towns around them.

In the 1950s, a local doctor told Stewart Wolf, a medical researcher, that hardly anyone in Roseto under age sixty-five had heart disease. Wolf was curious. He did blood draws and EKG scans, and he analyzed physicians’ records and death certificates. Wolf was amazed that almost no one under fifty-five showed any signs of heart disease, which at the time was America’s leading cause of death in men under sixty-five. Wolf enlisted the help of John Bruhn, a sociologist. They found that there was no suicide, alcoholism, or drug addiction in Roseto: everyone seemed to be dying of old age. According to Gladwell, Roseto was an outlier.

Section 2. ​

Wolf was forced to rule out diet and lifestyle as explanations, because the Rosetans had poor eating habits, smoked heavily, and were obese. He ruled out genetics after determining that Rosetans’ relatives in other parts of the country were not in comparably good health. Wolf eventually decided that what made Roseto’s inhabitants so healthy was the town itself—its culture and social structure. Many homes contained three generations living together, and with a population of just two thousand people, there were twenty-two separate civic organizations.

Wolf and Bruhn were met with a great deal of skepticism. They had to work hard to convince the medical establishment that the values of where we live and the people around us have a significant impact on who we are. Gladwell wants Outliers to do for our understanding of success what Wolf did for our understanding of health.

Summary: Chapter One: The Matthew Effect ​

Section 1. ​

Gladwell describes a game between two teams in the Canadian Hockey League, the most competitive junior hockey league in the world. Starting at a very young age, players are regularly assessed for talent, so that the best can be separated out and prepared for the next level. The system is designed to be a meritocracy.

Section 2. ​

Gladwell wants to show that circumstances matter just as much for success as work ethic or intelligence. People often ask about the lifestyle, intelligence, or special talents of successful individuals, but not enough people ask when and where a person grew up. Gladwell likens people to trees: we all know that the tallest trees received the most sunlight, had nutrient-rich soil, and were lucky enough to avoid lumberjacks. He suggests that we study the environmental and social influences that make people successful.

Section 3. ​

In the 1980s, Canadian psychologist Roger Barnsley observed that on elite Canadian hockey teams, whether youth league or NHL, a disproportionate number of players have birthdays in the first three months of the year.

Section 4. ​

The explanation for the distribution of birthdays is simple: the cutoff date for an individual’s league assignment is January 1. A player born on January 2 will be 12 months older than one born at the end of December. At the age of nine or ten, this often translates into a huge difference in size, coordination, and maturity. By age ten, coaches regularly pick the oldest and largest potential players for the more competitive “rep squads,” where the players will receive better coaching, have more practice time, and play many more games. By thirteen or fourteen, all of the extra experience and coaching will have actually made the players better, and they will be selected for the more competitive league(s). The same thing happens in American youth baseball and European soccer, but with different cutoff dates.

The phenomenon is also observable in education. Economists Kelly Bedard and Elizabeth Dhuey found that on math and science tests, the older children in fourth grade regularly score better than the youngest children. This can affect whether a child qualifies for a gifted program. When Bedard and Dhuey extended their analysis to four-year colleges, they found that the students who belonged to the relatively youngest group were underrepresented by more than 10 percent.

Section 5. ​

The top-level hockey players were given opportunities they neither deserved nor earned. Sociologist Robert Merton called this the “Matthew Effect,” after the verse in the Gospel of Matthew:

“For unto everyone that hath shall be given, and he shall have abundance. But from that hath not shall be taken away even that which he hath.”

The successful receive more resources: the best students receive the best teaching, the richest citizens get the biggest tax breaks, the largest kids get the best coaching. Sociologists call this “accumulative advantage.” Gladwell points out that cutoff dates exclude nearly half of the potential athletes from a population. Junior hockey teams in the Czech Republic show the same unbalanced distribution of birthdays as in Canada. If Canada or the Czech Republic separated each of their youth leagues into two separate leagues, one for kids born in the first half of the year, and one for kids born in the second half of the year, they would have twice as many athletes available for their national teams. Similarly, schools could separate kindergarten classes into students born in the first, middle, and last four months of the year. The benefits would justify the added administrative work, but, writes Gladwell, the scheme would run afoul of society’s need to believe that individual merit matters more than the rules society has created.

Section 6. ​

Gord Wasden, the father of one of the Canadian Hockey League boys from the first section, describes his son. Scott Wasden worked hard to get where he is, but he has also generally had the advantage of being big compared to others on his team. He was born on January 4.

Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect ​

Gladwell's thesis argues that that the idea of rugged, individual success is not accurate. Rather, the most successful person doesn’t thrive without some environmental and social influence plus a dose of good fortune.  After laying out the concepts of a meritocracy and suggesting that this is the way that hockey players advance in Canada, he immediately undercuts this idea. Gladwell uses the illustration of birthday distribution as an example of an invisible contributor to the success of hockey players. With a registration cutoff date of January 1, players born from January 2 through March 31 are older than teammates born later in the year. These older players are likely bigger, faster, more coordinated, and more mature. Since the older players are seen as more skilled, they get selected for more advanced leagues with more practice, more games, and higher-quality coaching. By the time these older players reach their early teens, the extra advantages gained from being judged better because of their birthday translate into being better players. Similarly, the age cutoff for school registrations often mistakes maturity for ability. As a result, the older children appear smarter, get put into advanced groups, and qualify for gifted programs. The birthdate bias carries through to college.

Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that "success leads to more success." More simply: by being a little bit better, a hockey player will get opportunities that may result in the player becoming an outlier. Gladwell implies that the systems that determine success are not efficient. He asks readers to question the way we, as a society, think about success. He suggests that changing the systems in athletics and education would chip away at the myth of individual merit as the chief marker of success and help level the playing field. Gladwell's model for this phenomenon is hockey player Scott Wasden. By demonstrating that Scott had the passion, talent, and work ethic for the game, as well as the good luck to be born on January 4, Gladwell reasons that Wasden, like many others he will be profiling, benefit by being both good and lucky.

Reference

  • “Outliers Introduction & Chapter 1 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section1/.
+ \ No newline at end of file diff --git a/save/reading/outliers/2.html b/save/reading/outliers/2.html index 6124a646..cc82c04e 100644 --- a/save/reading/outliers/2.html +++ b/save/reading/outliers/2.html @@ -5,12 +5,12 @@ Chapter Two: The 10,000-Hour Rule | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Chapter Two: The 10,000-Hour Rule ​

Summary: Chapter Two: The 10,000-Hour Rule ​

Section 1. ​

Bill Joy, a co-founder of Sun Microsystems, is one of the most influential computer programmers of all time. Joy attended the University of Michigan, thinking to become a biologist or mathematician, but he became obsessed with the Computer Center that opened during his freshman year. In 1975, he enrolled in graduate school at UC Berkeley. At both schools, Bill Joy spent much of his free time programming. After graduating from Berkeley, he founded Sun Microsystems, eventually rewriting the Java programming language. He would appear to be a good example of success based on individual merit.

Section 2. ​

If achievement is the combination of talent and preparation, how much does innate talent affect one’s success? In the 1990s, a study at Berlin’s elite Academy of Music determined that, regardless of instrument, all of the world-class students had practiced more than 10,000 hours. They did not find anyone at the top level who spent less time, nor did they find anyone who spent that amount of time who was not successful. Neurologist Daniel Levitin states that, across many fields, world-class expertise is never accomplished in less than 10,000 hours. The law holds even for a prodigy like Mozart, who started composing at age six: he did not compose his first masterwork until he was twenty-one. The youth athletes from the previous chapter who made it to the professional level put in 10,000 hours of practice. And note: individuals cannot normally put in that much practice without help. Most people will need supportive parents, financial stability, and probably a special program—like an all-star youth hockey team.

Section 3. ​

Bill Joy was certainly intelligent, having achieved a perfect score on the math portion of his Scholastic Aptitude Test. His opportunities, however, were extraordinary. Bill Joy happened to choose a college that had a time-sharing computer available, something that was exceedingly rare during the early 1970s. While most students had to pay for computer time, Bill Joy exploited a bug in the college’s system to gain unlimited access to the Computer Center, which was walking distance from where he lived. He was able to spend almost all of his free time programming, and when he went to Berkeley, he had a terminal at home, where he could program until he fell asleep at his keyboard. Bill Joy was able to accumulate 10,000 hours while in college.

Section 4. ​

The Beatles are another example of the 10,000-hours principle. In 1960, when they were still a high school rock band, they were given the chance to travel to Hamburg, Germany and play in various strip clubs. The Beatles were contracted to play for hours every night, seven nights a week, and thereby were forced to develop as musicians. After a year and a half, they had performed 270 nights. By 1964, when they first found commercial success, they had performed an estimated twelve hundred times—more than most band bands do in an entire career.

Section 5. ​

Born in Seattle to affluent parents, Bill Gates was sent to a private school for seventh grade. The school’s Mother’s Club funded the purchase of a time-sharing terminal connected to a mainframe computer in Seattle. As an eighth-grader, Gates was able to start programming. When the Mother’s Club could no longer pay for computing time, Gates was able to continue programming at Information Sciences, Inc. (ISI) through a connection he made with one of the school parents. After that, one of ISI’s founders recommended Gates for a programming project at the Bonneville Power station. Gates was able to leave high school during his senior year to program for the Bonneville Power station as an independent study project. By the time Bill Gates dropped out of Harvard, he was well past 10,000 hours of experience.

Section 6. ​

In a list of the seventy-five richest people in human history, going as far back as ancient Egypt and converting all wealth estimates into modern U.S. dollars, fourteen names are Americans born within nine years of one another (including John D. Rockefeller and J.P. Morgan). What accounts for the fact that a list that covers thousands of years should have 20 percent of its members so close together? The fourteen Americans happened to live during the 1860s and 1870s, when the American economy was undergoing a huge transformation: the expansion of the railroads and the creation of Wall Street. Gladwell argues that the fourteen names are so close in birth date because they were all the perfect age to seize the opportunities available.

Bill Joy and Bill Gates were also born at just the right time. In 1975, Popular Electronics ran a cover story about the Altair 8800, a $397 microcomputer kit. Someone born before 1954 would likely already have a job with IBM, working on mainframes. Someone born after 1956 would likely still be in high school. Most of the biggest pioneers in the modern computer industry, including Bill Gates, Bill Joy, Steve Jobs, and Eric Schmidt, were born in 1954 or 1955, as were Joy’s three fellow founders at Sun.

Analysis: Chapter Two: The 10,000-Hour Rule ​

Gladwell introduces his controversial and widely disputed 10,000 hours rule. Simply put, the rule suggests that to master a skill, an individual needs to practice it for 10,000 hours. Gladwell argues that, like hockey, success in computing requires both skill and luck, rather than pure, simple, natural ability. Gladwell's example is situated at the University of Michigan, specifically the new computer center, in 1971. The story of Bill Joy undermines the idea of success as an inexplicable, miraculous accomplishment. Though Gladwell agrees Joy is bright and talented, Joy also lucks into several happy accidents that allow for increased access. These strokes of luck give Joy the opportunity to become a computer expert. Gladwell's use of a psychological study conducted at the Academy of Music in Berlin, Germany, in the early 1990s forms the crux of his argument in this chapter. However, critics have suggested that this study was flawed and that there is no incontrovertible evidence to suggest that doing anything for 10,000 hours guarantees mastery. Rather, practice is, as Gladwell points out elsewhere throughout the text, just one of the factors in success. However, he also correlates the familial and economic advantages necessary to reach 10,000 of practice time, reinforcing the larger thesis of the book.

Gladwell argues that the Beatles and Bill Gates are examples of his notion of time spent directly relating to mastery. While detailing how the Beatles reached the 10,000-hour threshold for mastery, Gladwell also highlights the lucky breaks they had in achieving success, like the connection that landed the band in Hamburg’s music scene. Additionally, while Gates has become something of an American folk hero, Gladwell argues that Gates did not reach his great achievements completely on his own. His wealthy family, his lucky interaction with a computer in 1968, and his college experience allowed him far greater access to opportunity. Without this combination of factors, along with his intelligence and drive, Gates would likely not have become a household name. Again, Gladwell reinforces the idea that success comes not only from being good, but from having no small amount of luck, whether that be time, money, or, in an echo of the hockey player example, birthdate.

Gladwell claims that one important factor of success is the year in which one is born. In terms of wealth, there are 14 Americans, born in a nine-year period of the 1800s, who are among the richest people in history. This suggests that there was some connection to when they were born and the opportunities available to them in terms of innovation and invention, as well as the position they were in to take advantage of the opportunities. Similarly, men born in the same era of Bill Gates were coming of age at the bleeding edge of technology. Proximity to social and technological change, as well as the wealth and connections to take advantage of it, put these men in an advantageous position. Being in the right place at the right time, Gladwell argues, has at least as much to do with success as one's inborn abilities.

Reference

  • “Outliers Chapter 2 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section2/. Accessed 27 Feb. 2023.
- +
Toshiki's Note

Appearance

TOC

Chapter Two: The 10,000-Hour Rule ​

Summary: Chapter Two: The 10,000-Hour Rule ​

Section 1. ​

Bill Joy, a co-founder of Sun Microsystems, is one of the most influential computer programmers of all time. Joy attended the University of Michigan, thinking to become a biologist or mathematician, but he became obsessed with the Computer Center that opened during his freshman year. In 1975, he enrolled in graduate school at UC Berkeley. At both schools, Bill Joy spent much of his free time programming. After graduating from Berkeley, he founded Sun Microsystems, eventually rewriting the Java programming language. He would appear to be a good example of success based on individual merit.

Section 2. ​

If achievement is the combination of talent and preparation, how much does innate talent affect one’s success? In the 1990s, a study at Berlin’s elite Academy of Music determined that, regardless of instrument, all of the world-class students had practiced more than 10,000 hours. They did not find anyone at the top level who spent less time, nor did they find anyone who spent that amount of time who was not successful. Neurologist Daniel Levitin states that, across many fields, world-class expertise is never accomplished in less than 10,000 hours. The law holds even for a prodigy like Mozart, who started composing at age six: he did not compose his first masterwork until he was twenty-one. The youth athletes from the previous chapter who made it to the professional level put in 10,000 hours of practice. And note: individuals cannot normally put in that much practice without help. Most people will need supportive parents, financial stability, and probably a special program—like an all-star youth hockey team.

Section 3. ​

Bill Joy was certainly intelligent, having achieved a perfect score on the math portion of his Scholastic Aptitude Test. His opportunities, however, were extraordinary. Bill Joy happened to choose a college that had a time-sharing computer available, something that was exceedingly rare during the early 1970s. While most students had to pay for computer time, Bill Joy exploited a bug in the college’s system to gain unlimited access to the Computer Center, which was walking distance from where he lived. He was able to spend almost all of his free time programming, and when he went to Berkeley, he had a terminal at home, where he could program until he fell asleep at his keyboard. Bill Joy was able to accumulate 10,000 hours while in college.

Section 4. ​

The Beatles are another example of the 10,000-hours principle. In 1960, when they were still a high school rock band, they were given the chance to travel to Hamburg, Germany and play in various strip clubs. The Beatles were contracted to play for hours every night, seven nights a week, and thereby were forced to develop as musicians. After a year and a half, they had performed 270 nights. By 1964, when they first found commercial success, they had performed an estimated twelve hundred times—more than most band bands do in an entire career.

Section 5. ​

Born in Seattle to affluent parents, Bill Gates was sent to a private school for seventh grade. The school’s Mother’s Club funded the purchase of a time-sharing terminal connected to a mainframe computer in Seattle. As an eighth-grader, Gates was able to start programming. When the Mother’s Club could no longer pay for computing time, Gates was able to continue programming at Information Sciences, Inc. (ISI) through a connection he made with one of the school parents. After that, one of ISI’s founders recommended Gates for a programming project at the Bonneville Power station. Gates was able to leave high school during his senior year to program for the Bonneville Power station as an independent study project. By the time Bill Gates dropped out of Harvard, he was well past 10,000 hours of experience.

Section 6. ​

In a list of the seventy-five richest people in human history, going as far back as ancient Egypt and converting all wealth estimates into modern U.S. dollars, fourteen names are Americans born within nine years of one another (including John D. Rockefeller and J.P. Morgan). What accounts for the fact that a list that covers thousands of years should have 20 percent of its members so close together? The fourteen Americans happened to live during the 1860s and 1870s, when the American economy was undergoing a huge transformation: the expansion of the railroads and the creation of Wall Street. Gladwell argues that the fourteen names are so close in birth date because they were all the perfect age to seize the opportunities available.

Bill Joy and Bill Gates were also born at just the right time. In 1975, Popular Electronics ran a cover story about the Altair 8800, a $397 microcomputer kit. Someone born before 1954 would likely already have a job with IBM, working on mainframes. Someone born after 1956 would likely still be in high school. Most of the biggest pioneers in the modern computer industry, including Bill Gates, Bill Joy, Steve Jobs, and Eric Schmidt, were born in 1954 or 1955, as were Joy’s three fellow founders at Sun.

Analysis: Chapter Two: The 10,000-Hour Rule ​

Gladwell introduces his controversial and widely disputed 10,000 hours rule. Simply put, the rule suggests that to master a skill, an individual needs to practice it for 10,000 hours. Gladwell argues that, like hockey, success in computing requires both skill and luck, rather than pure, simple, natural ability. Gladwell's example is situated at the University of Michigan, specifically the new computer center, in 1971. The story of Bill Joy undermines the idea of success as an inexplicable, miraculous accomplishment. Though Gladwell agrees Joy is bright and talented, Joy also lucks into several happy accidents that allow for increased access. These strokes of luck give Joy the opportunity to become a computer expert. Gladwell's use of a psychological study conducted at the Academy of Music in Berlin, Germany, in the early 1990s forms the crux of his argument in this chapter. However, critics have suggested that this study was flawed and that there is no incontrovertible evidence to suggest that doing anything for 10,000 hours guarantees mastery. Rather, practice is, as Gladwell points out elsewhere throughout the text, just one of the factors in success. However, he also correlates the familial and economic advantages necessary to reach 10,000 of practice time, reinforcing the larger thesis of the book.

Gladwell argues that the Beatles and Bill Gates are examples of his notion of time spent directly relating to mastery. While detailing how the Beatles reached the 10,000-hour threshold for mastery, Gladwell also highlights the lucky breaks they had in achieving success, like the connection that landed the band in Hamburg’s music scene. Additionally, while Gates has become something of an American folk hero, Gladwell argues that Gates did not reach his great achievements completely on his own. His wealthy family, his lucky interaction with a computer in 1968, and his college experience allowed him far greater access to opportunity. Without this combination of factors, along with his intelligence and drive, Gates would likely not have become a household name. Again, Gladwell reinforces the idea that success comes not only from being good, but from having no small amount of luck, whether that be time, money, or, in an echo of the hockey player example, birthdate.

Gladwell claims that one important factor of success is the year in which one is born. In terms of wealth, there are 14 Americans, born in a nine-year period of the 1800s, who are among the richest people in history. This suggests that there was some connection to when they were born and the opportunities available to them in terms of innovation and invention, as well as the position they were in to take advantage of the opportunities. Similarly, men born in the same era of Bill Gates were coming of age at the bleeding edge of technology. Proximity to social and technological change, as well as the wealth and connections to take advantage of it, put these men in an advantageous position. Being in the right place at the right time, Gladwell argues, has at least as much to do with success as one's inborn abilities.

Reference

  • “Outliers Chapter 2 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section2/. Accessed 27 Feb. 2023.
+ \ No newline at end of file diff --git a/save/reading/outliers/3.html b/save/reading/outliers/3.html index 9afa12b9..d140741e 100644 --- a/save/reading/outliers/3.html +++ b/save/reading/outliers/3.html @@ -5,12 +5,12 @@ Chapter 3: The Trouble with Geniuses, Part 1 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Chapter 3: The Trouble with Geniuses, Part 1 ​

Summary: Chapter Three: The Trouble with Geniuses, Part 1 ​

Section 1. ​

Chris Langan appeared on the television trivia show 1 vs. 100 after being the subject of interviews and at least one documentary, due to his extremely high IQ. He had little difficulty in high school, got a perfect score on his SAT, and read through Principia Mathematica as a teenager. On 1 vs. 100, Langan cashed out with winnings of $250,000.

Section 2. ​

Shortly after World War I, Lewis Terman, creator of the widely used Stanford-Binet IQ test, became fascinated with child prodigies. In 1921, he identified exceptionally gifted 1,470 students who came to be known as the “Termites.” Terman kept track of them over the years, carefully cataloging their educational achievement, marriages, psychological health, and employment. Terman was sure that the Termites would be the future elite of the United States. The same confidence underlies the use of IQ-like tests, such as the SAT, to assess applicants at Ivy League schools and corporations such as Microsoft and Google. Gladwell states that Terman was wrong about the Termites, and if he had met Chris Langan as a teenager, he would have been wrong about him, too.

Section 3. ​

Gladwell gives two examples—one easy and one much harder—of questions that use visual puzzles to test IQ, without requiring any language abilities or specific factual knowledge. The average human will score 100. Those scoring below 70 are considered mentally disabled, and those who are successful in graduate programs likely have an IQ of 115 or higher. According to Gladwell, however, above 120 IQ makes little or no difference to success. A similar point applies to basketball: height is certainly an advantage, but there is, at best, modest value in being six foot seven instead of six foot six. A basketball player just needs to be tall enough.

A list of the last twenty-five Americans to win a Nobel Prize in Medicine includes graduates of prestigious schools, such as Columbia, Harvard, and MIT, but also graduates of DePauw, Holy Cross, and Gettysburg College. Similarly, there have been American Nobel laureates in Chemistry from Harvard and MIT, but also from Notre Dame and the University of Illinois. Harvard might have more intelligent students, but many other colleges have students smart enough to win a Nobel.

At the University of Michigan law school, around 10 percent of students are racial minorities. Without affirmative action, there would be less than 3 percent. Looking at career performance after graduation, Michigan found no difference between white and minority graduates. Minority students come into law school with weaker academic credentials than their white counterparts, but they are above the threshold required to be just as successful.

Section 4. ​

Not all questions that test for intelligence have a single correct answer. A different sort of question might ask for a list of different uses for a brick, or a blanket. A question like this measures imagination and creativity. At a top British high school, one student lists more than ten uses, many of which are comical. Another student, a prodigy with one of the school’s highest IQs, lists just a few uses, all of which are practical. Compared to an IQ test, asking about different uses for a brick might be a better way to identify future Nobel Prize winners.

Section 5. ​

Terman’s error was using intelligence alone to choose his Termites. While some of the Termites had successful careers, an unexpectedly high went on to careers that even Terman didn’t consider to be successful. The two students tested by Terman’s team who actually did go on to win Nobels were not selected as Termites: their IQs were too low. Terman eventually concluded that “intellect and achievement are far from perfectly correlated.”

Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.

Analysis: Chapter Three: The Trouble with Geniuses, Part 1 ​

Intelligence, Gladwell argues, does not equal success. He dedicates this chapter to the examination of geniuses. Comparing the young subjects of Lewis Terman to Chris Langan, Gladwell argues against Terman’s idea that genius is a key to societal advancement. Gladwell lists Langan’s early accomplishments as a shining example of Terman's idea, then swiftly undercuts expectations. Gladwell argues again that individuals do not succeed or fail solely on their own, but rather due to a constellation of factors, many of which are uncontrollable. Langan's early achievements would suggest great things and a continued path toward accomplishment, but Gladwell includes the anecdote of his game show appearance to undermine the notion that genius equals success. Using two questions from an IQ test, Gladwell introduces the idea of an IQ threshold to examine the the larger notion of intelligence and its limitations. Arguing that the measure of someone’s intelligence is only helpful up to a point, Gladwell uses examples of schools that Nobel laureates attended for undergraduate studies. Some, he points out, are elite institutions; others are not. Additionally, he points to success among graduates of the law school at the University of Michigan. There was no quantifiable difference in success post-graduation between white and POC students, some of whom were admitted under affirmative action with slightly lower undergraduate grades and standardized test scores. Reasonably, then, the level of intelligence beyond a certain point is moot. All of the students were intelligent enough to graduate, and beyond that, he argues, their intelligence predicted nothing.

Elements other than intelligence must contribute to success. Gladwell expands on his notion of an intelligence threshold and examines these other factors through an example of a question on a divergence test. Using examples of two students at an elite British school, Gladwell shows that the divergence test has no right answers, but instead looks for answers that demonstrate uniqueness and creativity. Intelligence matters, but so does imagination, creativity, and outside-the-box thinking. Gladwell circles back to Terman, reinforcing that the error when choosing his Termites was a focus on intelligence alone. He once again turns his attention to Chris Langan, foreshadowing that Langan’s story did not follow a direct line to achievement. Intelligence alone does not offer a certain path to success.

Reference

  • “Outliers Chapter 3 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section3/. Accessed 27 Feb. 2023.
- +
Toshiki's Note

Appearance

TOC

Chapter 3: The Trouble with Geniuses, Part 1 ​

Summary: Chapter Three: The Trouble with Geniuses, Part 1 ​

Section 1. ​

Chris Langan appeared on the television trivia show 1 vs. 100 after being the subject of interviews and at least one documentary, due to his extremely high IQ. He had little difficulty in high school, got a perfect score on his SAT, and read through Principia Mathematica as a teenager. On 1 vs. 100, Langan cashed out with winnings of $250,000.

Section 2. ​

Shortly after World War I, Lewis Terman, creator of the widely used Stanford-Binet IQ test, became fascinated with child prodigies. In 1921, he identified exceptionally gifted 1,470 students who came to be known as the “Termites.” Terman kept track of them over the years, carefully cataloging their educational achievement, marriages, psychological health, and employment. Terman was sure that the Termites would be the future elite of the United States. The same confidence underlies the use of IQ-like tests, such as the SAT, to assess applicants at Ivy League schools and corporations such as Microsoft and Google. Gladwell states that Terman was wrong about the Termites, and if he had met Chris Langan as a teenager, he would have been wrong about him, too.

Section 3. ​

Gladwell gives two examples—one easy and one much harder—of questions that use visual puzzles to test IQ, without requiring any language abilities or specific factual knowledge. The average human will score 100. Those scoring below 70 are considered mentally disabled, and those who are successful in graduate programs likely have an IQ of 115 or higher. According to Gladwell, however, above 120 IQ makes little or no difference to success. A similar point applies to basketball: height is certainly an advantage, but there is, at best, modest value in being six foot seven instead of six foot six. A basketball player just needs to be tall enough.

A list of the last twenty-five Americans to win a Nobel Prize in Medicine includes graduates of prestigious schools, such as Columbia, Harvard, and MIT, but also graduates of DePauw, Holy Cross, and Gettysburg College. Similarly, there have been American Nobel laureates in Chemistry from Harvard and MIT, but also from Notre Dame and the University of Illinois. Harvard might have more intelligent students, but many other colleges have students smart enough to win a Nobel.

At the University of Michigan law school, around 10 percent of students are racial minorities. Without affirmative action, there would be less than 3 percent. Looking at career performance after graduation, Michigan found no difference between white and minority graduates. Minority students come into law school with weaker academic credentials than their white counterparts, but they are above the threshold required to be just as successful.

Section 4. ​

Not all questions that test for intelligence have a single correct answer. A different sort of question might ask for a list of different uses for a brick, or a blanket. A question like this measures imagination and creativity. At a top British high school, one student lists more than ten uses, many of which are comical. Another student, a prodigy with one of the school’s highest IQs, lists just a few uses, all of which are practical. Compared to an IQ test, asking about different uses for a brick might be a better way to identify future Nobel Prize winners.

Section 5. ​

Terman’s error was using intelligence alone to choose his Termites. While some of the Termites had successful careers, an unexpectedly high went on to careers that even Terman didn’t consider to be successful. The two students tested by Terman’s team who actually did go on to win Nobels were not selected as Termites: their IQs were too low. Terman eventually concluded that “intellect and achievement are far from perfectly correlated.”

Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.

Analysis: Chapter Three: The Trouble with Geniuses, Part 1 ​

Intelligence, Gladwell argues, does not equal success. He dedicates this chapter to the examination of geniuses. Comparing the young subjects of Lewis Terman to Chris Langan, Gladwell argues against Terman’s idea that genius is a key to societal advancement. Gladwell lists Langan’s early accomplishments as a shining example of Terman's idea, then swiftly undercuts expectations. Gladwell argues again that individuals do not succeed or fail solely on their own, but rather due to a constellation of factors, many of which are uncontrollable. Langan's early achievements would suggest great things and a continued path toward accomplishment, but Gladwell includes the anecdote of his game show appearance to undermine the notion that genius equals success. Using two questions from an IQ test, Gladwell introduces the idea of an IQ threshold to examine the the larger notion of intelligence and its limitations. Arguing that the measure of someone’s intelligence is only helpful up to a point, Gladwell uses examples of schools that Nobel laureates attended for undergraduate studies. Some, he points out, are elite institutions; others are not. Additionally, he points to success among graduates of the law school at the University of Michigan. There was no quantifiable difference in success post-graduation between white and POC students, some of whom were admitted under affirmative action with slightly lower undergraduate grades and standardized test scores. Reasonably, then, the level of intelligence beyond a certain point is moot. All of the students were intelligent enough to graduate, and beyond that, he argues, their intelligence predicted nothing.

Elements other than intelligence must contribute to success. Gladwell expands on his notion of an intelligence threshold and examines these other factors through an example of a question on a divergence test. Using examples of two students at an elite British school, Gladwell shows that the divergence test has no right answers, but instead looks for answers that demonstrate uniqueness and creativity. Intelligence matters, but so does imagination, creativity, and outside-the-box thinking. Gladwell circles back to Terman, reinforcing that the error when choosing his Termites was a focus on intelligence alone. He once again turns his attention to Chris Langan, foreshadowing that Langan’s story did not follow a direct line to achievement. Intelligence alone does not offer a certain path to success.

Reference

  • “Outliers Chapter 3 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section3/. Accessed 27 Feb. 2023.
+ \ No newline at end of file diff --git a/save/reading/outliers/4.html b/save/reading/outliers/4.html index 9d3c1267..3ecf1eae 100644 --- a/save/reading/outliers/4.html +++ b/save/reading/outliers/4.html @@ -5,12 +5,12 @@ Chapter 4: The Trouble with Geniuses, Part 2 | Toshiki's Note - + - + @@ -32,9 +32,9 @@ -
Toshiki's Note

Appearance

TOC

Chapter 4: The Trouble with Geniuses, Part 2 ​

Summary: Chapter Four: The Trouble with Geniuses, Part 2 ​

Section 1. ​

Chris Langan’s mother had four children, all from different fathers. Her first husband died in Mexico, the second was murdered, the third committed suicide, and the fourth was an abusive alcoholic. Langan says he has never met someone who grew up as impoverished as he and his brothers did. Attending Reed College, in Oregon, on a scholarship, he felt as though he did not belong with the rest of the students. When his mother neglected to fill out the financial aid forms, he lost his scholarship and dropped out. According to Langan, the school just did not care about their students. Later, Langan attended Montana State University in Bozeman but he dropped out when a dean refused Langan’s request to adjust his class schedule because of car trouble. He still has intellectual interests and has been working on a treatise called the “Cognitive Theoretic Model of the Universe.” When asked if, hypothetically, he would take a position at Harvard, he acknowledges the benefit of an environment with so much intellectual energy, but he worries about a lack of intellectual freedom.

Section 2. ​

Gladwell points out the oddity of Langan’s experiences. Most colleges, especially small ones, try to accommodate student needs. Smart people take positions at places like Harvard instead of in the private sector precisely for the added intellectual freedom. Gladwell contrasts Langan’s story with Robert Oppenheimer’s. While at Cambridge, Oppenheimer became frustrated with his tutor and tried to poison him. After negotiations with the school administration, Oppenheimer was put on probation and assigned to a psychiatrist. Whereas Langan dropped out after his mother failed to fill out paperwork, Oppenheimer managed to stay in college after trying to kill someone. Later, Oppenheimer convinced General Leslie Groves to make Oppenheimer the scientific director of the Manhattan Project. Oppenheimer had the ability to persuade people, where Langan did not.

Section 3. ​

Psychologist Robert Sternberg calls the ability to persuade someone “practical intelligence.” It involves knowing what to say, how to say it, and when. It requires an ability to read situations. Individuals can have either practical or analytical intelligence, or in the case of Oppenheimer, both. To understand where one obtains practical intelligence, Gladwell refers to a study conducted by Annette Lareau.

Lareau observed parenting strategies in both Black and white families from different socioeconomic strata. She found two distinct strategies: “concerted cultivation” and “accomplishment of natural growth.” Concerted cultivation, most common in middle-class and wealthier families, is defined by parents’ active promotion of their children’s talents and interests. Accomplishment of natural growth, most common in working-class or impoverished families, is defined by parents’ lack of interest or interaction in their children’s free time. Concerted cultivation, writes Gladwell, encourages children to seek opportunities for growth and to advocate for themselves. Accomplishment of natural growth leaves children unprepared to shape their environment for their own benefit. Concerted cultivation fosters an attitude better suited for success in the modern world.

Section 4. ​

Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.

  • Affluent: Having an abundant supply of money or possessions of value.

By contrast, Chris Langan’s brother Mark states that as children, they learned to resent authority. Mark, too, was unable to obtain financial aid, because he had no knowledge of how the system worked. If Chris Langan had been raised in a family that valued education, he probably would have had more success.

Section 5. ​

Once the Termites reached working adulthood, Terman grouped 730 of the males into three categories, based on their levels of success. The top 20 percent were upper-class professionals, while the bottom 20 percent had low-paying jobs or were unemployed. After analyzing the differences between the groups, Terman found that the bottom 20 percent came almost entirely from the lowest social and economic classes. What they lacked was “a community around them that prepared them properly for the world.”

Section 6. ​

Chris Langan lives on a horse farm in Missouri, where he still writes. He is happily married and continues to read physics and philosophy. While he has spent decades on his writing, almost none of it has ever been published. Langan admits he has not tried to contact publishers or found an agent—and does not plan to. Chris Langan did not have the support necessary to meet success. Gladwell points that no one ever makes it alone, regardless of if they are successful musicians, athletes, tech billionaires, or geniuses.

Analysis: Chapter Four: The Trouble with Geniuses, Part 2 ​

Family background plays a crucial role in an individual’s success. While a genius like Chris Langan should thrive on a college campus, he drops out. Langan doesn’t struggle with academics, but with paperwork, procedures, and environment. Gladwell’s characterization of Langan almost makes him seem like two separate characters: a confident, genius game show contestant versus a child from a poor and troubled family. Langan may be a true outlier in the sense that he doesn’t seem to fit in academia or the working-class world.  Langan’s story is ironic. His brilliant mind should propel him to success. However, his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success. The author delves deeper into the irony of Langan’s situation by comparing him to Robert Oppenheimer, who worked on the nuclear bomb during World War II. Like Langan, Oppenheimer possesses great intelligence. Unlike Langan, Oppenheimer has an understanding that allows him to navigate the world effectively. Oppenheimer knows how to combat challenges, where Langan gets bested by a financial aid problem and class schedule. Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority. Without a family background that encourages success, intelligence is not enough.

Contrast & Comparison

Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.

The power of persuasion is an important factor that drives success. Gladwell notes that the power of persuasion is practical intelligence. He contrasts practical intelligence with analytical ability, noting that just because a person is smart does not mean they have practical intelligence. These two types of knowledge are not necessarily coeval. He reinforces this point with a return to the comparison of Langan and Oppenheimer. Both men are exceedingly intelligent, but Langan lacks the social savvy of practical intelligence. Gladwell suggests that individuals gain practical intelligence skills from their families, discussing a study by sociologist Annette Lareau to prove his point. Children from wealthier families learn how to interact with society in a way that drives achievement. Successful people have the power of persuasion and can convince others to see things from their point of view.

Success, then, depends on more than individual effort. Gladwell builds this argument by detailing the differences between Oppenheimer’s upbringing and Langan’s. Oppenheimer lived in an upper-class neighborhood with myriad advantages. He learned how to negotiate and stand up for himself. Langan’s family moved from place to place, struggling for jobs, money, food, and clothes, and Langan learned that he couldn’t rely on others. Gladwell's consideration of the Termites' success or struggles as adults supports his assertion that success is more than an individual effort; Gladwell states, “In the end, only one thing mattered: family background.” Terman’s findings and Gladwell’s conclusion do not mean that talented individuals from lower social and economic classes cannot have successful futures. To succeed, everyone needs help along the way, if not from family, then from a supportive community. Individuals do not achieve success solely on their own.

Reference

  • “Outliers Chapter 4 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section4/.
- +
Toshiki's Note

Appearance

TOC

Chapter 4: The Trouble with Geniuses, Part 2 ​

Summary: Chapter Four: The Trouble with Geniuses, Part 2 ​

Section 1. ​

Chris Langan’s mother had four children, all from different fathers. Her first husband died in Mexico, the second was murdered, the third committed suicide, and the fourth was an abusive alcoholic. Langan says he has never met someone who grew up as impoverished as he and his brothers did. Attending Reed College, in Oregon, on a scholarship, he felt as though he did not belong with the rest of the students. When his mother neglected to fill out the financial aid forms, he lost his scholarship and dropped out. According to Langan, the school just did not care about their students. Later, Langan attended Montana State University in Bozeman but he dropped out when a dean refused Langan’s request to adjust his class schedule because of car trouble. He still has intellectual interests and has been working on a treatise called the “Cognitive Theoretic Model of the Universe.” When asked if, hypothetically, he would take a position at Harvard, he acknowledges the benefit of an environment with so much intellectual energy, but he worries about a lack of intellectual freedom.

Section 2. ​

Gladwell points out the oddity of Langan’s experiences. Most colleges, especially small ones, try to accommodate student needs. Smart people take positions at places like Harvard instead of in the private sector precisely for the added intellectual freedom. Gladwell contrasts Langan’s story with Robert Oppenheimer’s. While at Cambridge, Oppenheimer became frustrated with his tutor and tried to poison him. After negotiations with the school administration, Oppenheimer was put on probation and assigned to a psychiatrist. Whereas Langan dropped out after his mother failed to fill out paperwork, Oppenheimer managed to stay in college after trying to kill someone. Later, Oppenheimer convinced General Leslie Groves to make Oppenheimer the scientific director of the Manhattan Project. Oppenheimer had the ability to persuade people, where Langan did not.

Section 3. ​

Psychologist Robert Sternberg calls the ability to persuade someone “practical intelligence.” It involves knowing what to say, how to say it, and when. It requires an ability to read situations. Individuals can have either practical or analytical intelligence, or in the case of Oppenheimer, both. To understand where one obtains practical intelligence, Gladwell refers to a study conducted by Annette Lareau.

Lareau observed parenting strategies in both Black and white families from different socioeconomic strata. She found two distinct strategies: “concerted cultivation” and “accomplishment of natural growth.” Concerted cultivation, most common in middle-class and wealthier families, is defined by parents’ active promotion of their children’s talents and interests. Accomplishment of natural growth, most common in working-class or impoverished families, is defined by parents’ lack of interest or interaction in their children’s free time. Concerted cultivation, writes Gladwell, encourages children to seek opportunities for growth and to advocate for themselves. Accomplishment of natural growth leaves children unprepared to shape their environment for their own benefit. Concerted cultivation fosters an attitude better suited for success in the modern world.

Section 4. ​

Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.

  • Affluent: Having an abundant supply of money or possessions of value.

By contrast, Chris Langan’s brother Mark states that as children, they learned to resent authority. Mark, too, was unable to obtain financial aid, because he had no knowledge of how the system worked. If Chris Langan had been raised in a family that valued education, he probably would have had more success.

Section 5. ​

Once the Termites reached working adulthood, Terman grouped 730 of the males into three categories, based on their levels of success. The top 20 percent were upper-class professionals, while the bottom 20 percent had low-paying jobs or were unemployed. After analyzing the differences between the groups, Terman found that the bottom 20 percent came almost entirely from the lowest social and economic classes. What they lacked was “a community around them that prepared them properly for the world.”

Section 6. ​

Chris Langan lives on a horse farm in Missouri, where he still writes. He is happily married and continues to read physics and philosophy. While he has spent decades on his writing, almost none of it has ever been published. Langan admits he has not tried to contact publishers or found an agent—and does not plan to. Chris Langan did not have the support necessary to meet success. Gladwell points that no one ever makes it alone, regardless of if they are successful musicians, athletes, tech billionaires, or geniuses.

Analysis: Chapter Four: The Trouble with Geniuses, Part 2 ​

Family background plays a crucial role in an individual’s success. While a genius like Chris Langan should thrive on a college campus, he drops out. Langan doesn’t struggle with academics, but with paperwork, procedures, and environment. Gladwell’s characterization of Langan almost makes him seem like two separate characters: a confident, genius game show contestant versus a child from a poor and troubled family. Langan may be a true outlier in the sense that he doesn’t seem to fit in academia or the working-class world.  Langan’s story is ironic. His brilliant mind should propel him to success. However, his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success. The author delves deeper into the irony of Langan’s situation by comparing him to Robert Oppenheimer, who worked on the nuclear bomb during World War II. Like Langan, Oppenheimer possesses great intelligence. Unlike Langan, Oppenheimer has an understanding that allows him to navigate the world effectively. Oppenheimer knows how to combat challenges, where Langan gets bested by a financial aid problem and class schedule. Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority. Without a family background that encourages success, intelligence is not enough.

Contrast & Comparison

Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.

The power of persuasion is an important factor that drives success. Gladwell notes that the power of persuasion is practical intelligence. He contrasts practical intelligence with analytical ability, noting that just because a person is smart does not mean they have practical intelligence. These two types of knowledge are not necessarily coeval. He reinforces this point with a return to the comparison of Langan and Oppenheimer. Both men are exceedingly intelligent, but Langan lacks the social savvy of practical intelligence. Gladwell suggests that individuals gain practical intelligence skills from their families, discussing a study by sociologist Annette Lareau to prove his point. Children from wealthier families learn how to interact with society in a way that drives achievement. Successful people have the power of persuasion and can convince others to see things from their point of view.

Success, then, depends on more than individual effort. Gladwell builds this argument by detailing the differences between Oppenheimer’s upbringing and Langan’s. Oppenheimer lived in an upper-class neighborhood with myriad advantages. He learned how to negotiate and stand up for himself. Langan’s family moved from place to place, struggling for jobs, money, food, and clothes, and Langan learned that he couldn’t rely on others. Gladwell's consideration of the Termites' success or struggles as adults supports his assertion that success is more than an individual effort; Gladwell states, “In the end, only one thing mattered: family background.” Terman’s findings and Gladwell’s conclusion do not mean that talented individuals from lower social and economic classes cannot have successful futures. To succeed, everyone needs help along the way, if not from family, then from a supportive community. Individuals do not achieve success solely on their own.

Reference

  • “Outliers Chapter 4 Summary & Analysis.” SparkNotes, www.sparknotes.com/lit/outliers/section4/.
+ \ No newline at end of file diff --git a/sitemap.xml b/sitemap.xml index 44c245f5..d9f2ee0e 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -1 +1 @@ -https://note.toshiki.dev/getting-started2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/roadmap2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/save/reading/outliers/32023-05-23T16:22:25.000Zhttps://note.toshiki.dev/application/markdown-it-katex/support-function2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/notes/12-52023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/literature/writing/methods-of-development2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/12023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/README2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/save/reading/outliers/42023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/problems/02-202023-05-23T16:22:25.000Zhttps://note.toshiki.dev/application/markdown-it-katex/how-to-use2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/save/reading/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/save/reading/outliers/12023-05-23T16:22:25.000Zhttps://note.toshiki.dev/application/markdown-it-katex/support-table2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-12023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-22023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-32023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/literature/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/102023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/112023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/122023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/132023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/32023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/22023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/42023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/52023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/72023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/62023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/82023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/92023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/vocabulary/2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/academic/vocabulary/2023/02/2023-02-272023-05-23T16:22:25.000Zhttps://note.toshiki.dev/application/markdown-it-katex/tips2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/acknowledgement2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/assignments2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/author2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/closing2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/license2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/javascript/notes/1/1-12023-05-23T16:22:25.000Zhttps://note.toshiki.dev/save/reading/outliers/22023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/appendix2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/aws-batch2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/aws-get-started2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/cloud2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/docker-system2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/handson-bashoutter2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/handson-ec22023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/handson-jupyter2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/handson-qabot2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/handson-serverless2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/introduction2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/scientific-computing2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/serverless2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/webserver2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/development/aws/main2023-05-23T16:22:25.000Zhttps://note.toshiki.dev/javascript/notes/1/1-22023-05-23T16:22:25.000Z \ No newline at end of file +https://note.toshiki.dev/development/file-naming-convention2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/rclone-for-r22023-05-23T16:32:31.000Zhttps://note.toshiki.dev/getting-started2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/roadmap2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/application/markdown-it-katex/support-table2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/notes/12-52023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/literature/writing/methods-of-development2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/12023-05-23T16:32:31.000Zhttps://note.toshiki.dev/application/markdown-it-katex/support-function2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-32023-05-23T16:32:31.000Zhttps://note.toshiki.dev/application/markdown-it-katex/how-to-use2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/README2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/save/reading/outliers/12023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/literature/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/102023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/112023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/122023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/132023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/22023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/32023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/42023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/52023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/62023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/72023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/82023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/physics/ipho-formulas-jpn/92023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/vocabulary/2023/02/2023-02-272023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/vocabulary/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/application/markdown-it-katex/tips2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/acknowledgement2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/assignments2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/author2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/closing2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/license2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/javascript/notes/1/1-12023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/problems/02-202023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-12023-05-23T16:32:31.000Zhttps://note.toshiki.dev/academic/chemistry/problems/03-02-22023-05-23T16:32:31.000Zhttps://note.toshiki.dev/save/reading/2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/save/reading/outliers/22023-05-23T16:32:31.000Zhttps://note.toshiki.dev/save/reading/outliers/32023-05-23T16:32:31.000Zhttps://note.toshiki.dev/save/reading/outliers/42023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/appendix2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/aws-batch2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/aws-get-started2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/cloud2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/docker-system2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/handson-bashoutter2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/handson-ec22023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/handson-jupyter2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/handson-qabot2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/handson-serverless2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/introduction2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/scientific-computing2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/serverless2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/main2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/development/aws/webserver2023-05-23T16:32:31.000Zhttps://note.toshiki.dev/javascript/notes/1/1-22023-05-23T16:32:31.000Z \ No newline at end of file